12 PRE-NUMBERED NEW SEQUENCES A079216 - A079227 follow: (I will later compute the corresponding exclusive variant with the Moebius transformation...) %I A079216 %S A079216 1,1,1,2,1,1,3,2,1,1,5,5,2,1,1,6,11,3,2,1,1,10,26,8,5,2,1,1,11,66,18,11,3,2,1,1,18,161,43,30, %T A079216 5,5,2,1,1,21,420,104,82,6,14,3,2,1,1,34,1093,273,233,15,38,5,5,2,1,1,35,2916,702,680,36,111, %U A079216 6,11,3,2,1,1,68,7819,1870,2033,108,332,10,30,8,5,2,1,1,69,21304,4985,6164,301,1029,11,82,18 %N A079216 Square array A(n>=0,k>=1) (listed antidiagonally: A(0,1)=1, A(1,1)=1, A(0,2)=1, A(2,1)=2, A(1,2)=1, A(0,3)=1, A(3,1)=3, ...) giving the number of n-edge general plane trees fixed by k-fold application of the gatomorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees). %C A079216 Note: the counts given here are inclusive, e.g. A(n,6) includes the counts A(n,3) and A(n,2) which in turn both include A(n,1). %F A079216 A(0,k) = 1. A(n,k) = Sum_{r=1..n where r/gcd(r,k) divides n} Sum_{c as each composition of n/(r/gcd(r,k)) into gcd(r,k) parts} Product_{i as each composant of c} A(i-1,lcm(r,k)) %R A079216 %H A079216 A. Karttunen, Gatomorphisms %O A079216 0,4 %K A079216 nonn,tabl,new %A A079216 Antti Karttunen (my_firstname.my_surname@iki.fi) Jan 03 2002 %Y A079216 A(n,A003418(n)) = A000108(n). The first row: A057546, second: A079223, third: A079224, fourth: A079225, fifth: A079226, sixth: A079227. Cf. also A079217-A079222. %D A079216 %p A079216 with(combinat,composition); # composition(n,k) gives ordered partitions of integer n into k parts. %p A079216 [seq(A079216(n),n=0..119)]; A079216 := n -> A079216bi(A025581(n),A002262(n)+1); %p A079216 A079216bi :=proc(n,k) option remember; local r; if(0 = n) then RETURN(1); else RETURN(add(PFixedByA057511(n,k,r),r=1..n)); fi; end; %p A079216 PFixedByA057511 :=proc(n,k,r) option remember; local ncycles, cyclen, i, c; ncycles := igcd(r,k); cyclen := r/ncycles; if(0 <> (n mod cyclen)) then RETURN(0); else add(mul(A079216bi(i-1,ilcm(r,k)),i=c),c=composition(n/cyclen,ncycles)); fi; end; %I A079217 %S A079217 1,1,1,2,0,1,3,1,0,1,5,0,0,0,1,6,2,1,0,0,1,10,0,0,0,0,0,1,11,5,0,1,0,0,0,1,18,0,2,0, %T A079217 0,0,0,0,1,21,11,0,0,1,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,1,35,26,3,2,0,1,0,0,0,0,0,1,68,0, %U A079217 0,0,0,0,0,0,0,0,0,0,1,69,66,0,0,0,0,1,0,0,0,0,0,0,1,137,0,8,0,2,0,0,0,0,0,0,0,0,0,1 %N A079217 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=0, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by the gatomorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees). %R A079217 %O A079217 0,4 %K A079217 nonn,tabl,new %A A079217 Antti Karttunen (my_firstname.my_surname@iki.fi) Jan 03 2002 %Y A079217 The row sums equal to the left edge shifted left once = A057546 = first row of A079216 (the latter gives the Maple procedure PFixedByA057511). Cf. also A079218, A079219, A079220, A079221, A079222 and A003056 & A002262. %D A079217 %p A079217 [seq(A079217(n),n=0..119)]; A079217 := n -> PFixedByA057511(A003056(n)+1,1,A002262(n)+1); %I A079218 %S A079218 1,1,1,2,2,1,5,5,0,1,11,14,0,0,1,26,36,1,2,0,1,66,94,0,0,0,0,1,161,253,0,5,0,0,0,1,420,668, %T A079218 2,0,0,2,0,0,1,1093,1807,0,14,1,0,0,0,0,1,2916,4902,0,0,0,0,0,0,0,0,1,7819,13436,5,36,0,5,0, %U A079218 2,0,0,0,1,21304,37016,0,0,0,0,0,0,0,0,0,0,1,58321,102808,0,102,0,0,1,0,0,0,0,0,0,1,161233 %N A079218 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=2, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by the two-fold application of gatomorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees). %C A079218 Note: the counts given here are inclusive, i.e. T(n,d) includes also the count A079217(n,d). %R A079218 %O A079218 0,4 %K A079218 nonn,tabl,new %A A079218 Antti Karttunen (my_firstname.my_surname@iki.fi) Jan 03 2002 %Y A079218 The row sums equal to the left edge shifted left once = A079223 = second row of A079216 (the latter gives the Maple procedure PFixedByA057511). Cf. also A079217-A079222 and A003056 & A002262. %D A079218 %p A079218 [seq(A079218(n),n=0..119)]; A079218 := n -> PFixedByA057511(A003056(n)+1,2,A002262(n)+1); %I A079219 %S A079219 1,1,1,2,0,1,3,1,3,1,8,0,9,0,1,18,2,22,0,0,1,43,0,60,0,0,0,1,104,5,159,1,0,3,0,1,273,0,428, %T A079219 0,0,0,0,0,1,702,14,1143,0,1,9,0,0,0,1,1870,0,3114,0,0,0,0,0,0,0,1,4985,38,8505,2,0,28,0,0, %U A079219 3,0,0,1,13562,0,23475,0,0,0,0,0,0,0,0,0,1,37038,111,65025,0,0,90,1,0,0,0,0,0,0,1,102266,0 %N A079219 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=0, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by the three-fold application of gatomorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees). %C A079219 Note: the counts given here are inclusive, i.e. T(n,d) includes also the count A079217(n,d). %R A079219 %O A079219 0,4 %K A079219 nonn,tabl,new %A A079219 Antti Karttunen (my_firstname.my_surname@iki.fi) Jan 03 2002 %Y A079219 The row sums equal to the left edge shifted left once = A079224 = third row of A079216 (the latter gives the Maple procedure PFixedByA057511). Cf. also A079217-A079222 and A003056 & A002262. %D A079219 %p A079219 [seq(A079219(n),n=0..119)]; A079219 := n -> PFixedByA057511(A003056(n)+1,3,A002262(n)+1); %I A079220 %S A079220 1,1,1,2,2,1,5,5,0,1,11,14,0,4,1,30,36,1,14,0,1,82,102,0,48,0,0,1,233,293,0,153,0,0,0,1,680, %T A079220 860,2,488,0,2,0,0,1,2033,2575,0,1550,1,0,0,4,0,1,6164,7838,0,4920,0,0,0,0,0,0,1,18923,24148, %U A079220 5,15672,0,5,0,14,0,0,0,1,58768,75152,0,50124,0,0,0,0,0,0,0,0,1,184045,236032,0,160978,0,0 %N A079220 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=2, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by the four-fold application of gatomorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees). %C A079220 Note: the counts given here are inclusive, i.e. T(n,d) includes also the count A079218(n,d). %R A079220 %O A079220 0,4 %K A079220 nonn,tabl,new %A A079220 Antti Karttunen (my_firstname.my_surname@iki.fi) Jan 03 2002 %Y A079220 The row sums equal to the left edge shifted left once = A079225 = fourth row of A079216 (the latter gives the Maple procedure PFixedByA057511). Cf. also A079217-A079222 and A003056 & A002262. %D A079220 %p A079220 [seq(A079220(n),n=0..119)]; A079220 := n -> PFixedByA057511(A003056(n)+1,4,A002262(n)+1); %I A079221 %S A079221 1,1,1,2,0,1,3,1,0,1,5,0,0,0,1,6,2,1,0,5,1,15,0,0,0,20,0,1,36,5,0,1,65,0,0,1,108,0,2,0, %T A079221 190,0,0,0,1,301,11,0,0,501,0,0,0,0,1,814,0,0,0,1265,0,0,0,0,0,1,2080,26,3,2,3105,1,0,0,0,5, %U A079221 0,1,5223,0,0,0,7695,0,0,0,0,0,0,0,1,12919,71,0,0,19545,0,1,0,0,20,0,0,0,1,32557,0,8,0,51377 %N A079221 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=0, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by the five-fold application of gatomorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees). %C A079221 Note: the counts given here are inclusive, i.e. T(n,d) includes also the count A079217(n,d). %R A079221 %O A079221 0,4 %K A079221 nonn,tabl,new %A A079221 Antti Karttunen (my_firstname.my_surname@iki.fi) Jan 03 2002 %Y A079221 The row sums equal to the left edge shifted left once = A079226 = fifth row of A079216 (the latter gives the Maple procedure PFixedByA057511). Cf. also A079217-A079222 and A003056 & A002262. %D A079221 %p A079221 [seq(A079221(n),n=0..119)]; A079221 := n -> PFixedByA057511(A003056(n)+1,5,A002262(n)+1); %I A079222 %S A079222 1,1,1,2,2,1,5,5,3,1,14,14,9,0,1,38,42,28,2,0,1,111,124,90,0,0,6,1,332,379,285,5,0,27,0,1,1029, %T A079222 1178,914,0,0,110,0,0,1,3232,3742,2955,14,1,429,0,0,0,1,10374,12024,9666,0,0,1614,0,0,0,0,1, %U A079222 33679,39200,31853,42,0,5942,0,2,3,0,0,1,110722,129056,105909,0,0,21564,0,0,0,0,0,0,1,367252 %N A079222 %N A079222 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=2, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by the six-fold application of gatomorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees). %C A079222 Note: the counts given here are inclusive, i.e. T(n,d) includes also the counts A079218(n,d) and A079219(n,d). %R A079222 %O A079222 0,4 %K A079222 nonn,tabl,new %A A079222 Antti Karttunen (my_firstname.my_surname@iki.fi) Jan 03 2002 %Y A079222 The row sums equal to the left edge shifted left once = A079227 = sixth row of A079216 (the latter gives the Maple procedure PFixedByA057511). Cf. also A079217-A079221 and A003056 & A002262. %D A079222 %p A079222 [seq(A079222(n),n=0..119)]; A079222 := n -> PFixedByA057511(A003056(n)+1,6,A002262(n)+1); %I A079223 %S A079223 1,1,2,5,11,26,66,161,420,1093,2916,7819,21304,58321,161233,448090,1253252,3521389,9941693, %T A079223 28175716,80152141,228747967,654817275,1879602446,5408974390,15601662378,45098766532,130624550412, %U A079223 379047234158,1101829730685,3208043336588,9354609854849,27316870741493,79876524974454,233860548401666 %N A079223 Number of Catalan objects fixed by two-fold application of the gatomorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees). %F A079223 a(n) = A079216(n,2) %R A079223 %O A079223 0,3 %K A079223 nonn,new %A A079223 Antti Karttunen (my_firstname.my_surname@iki.fi) Jan 03 2002 %Y A079223 The second row of A079216. The leftmost edge of the triangle A079218 and also its row sums shifted by one. Occurs for first time in A073202 as row 245. Cf. A057546, A079224, A079225, A079226, A079227. %D A079223 %p A079223 A079223 := n -> A079216bi(n,2); %I A079224 %S A079224 1,1,2,3,8,18,43,104,273,702,1870,4985,13562,37038,102266,283774,793189,2227115,6286044,17811751, %T A079224 50672898,144639235,414181050,1189365940,3424477813,9883578364,28589660227,82870288432,240672107114, %U A079224 700211561751,2040611192500,5956223203778,17410858226139,50964656080528,149376798368372,438359238488252 %N A079224 Number of Catalan objects fixed by three-fold application of the gatomorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees). %F A079224 a(n) = A079216(n,3) %R A079224 %O A079224 0,3 %K A079224 nonn,new %A A079224 Antti Karttunen (my_firstname.my_surname@iki.fi) Jan 03 2002 %Y A079224 The third row of A079216. The leftmost edge of the triangle A079219 and also its row sums shifted by one. Occurs in A073202 as row 43639. Cf. A057546, A079223-A079227. %D A079224 %p A079224 A079224 := n -> A079216bi(n,3); %I A079225 %S A079225 1,1,2,5,11,30,82,233,680,2033,6164,18923,58768,184045,581105,1846906,5905364,18980465,61292929, %T A079225 198758704,646974285,2113163707,6923642271,22749608810,74946337830,247499313730,819154110660, %U A079225 2716779932308,9027706698022,30052304981861,100208463554740,334665574064089,1119323838935653 %N A079225 Number of Catalan objects fixed by four-fold application of the gatomorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees). %F A079225 a(n) = A079216(n,4) %R A079225 %O A079225 0,3 %K A079225 nonn,new %A A079225 Antti Karttunen (my_firstname.my_surname@iki.fi) Jan 03 2002 %Y A079225 The fourth row of A079216. The leftmost edge of the triangle A079220 and also its row sums shifted by one. Occurs in A073202 as row 2290625151. Cf. A057546, A079223-A079227. %D A079225 %p A079225 A079225 := n -> A079216bi(n,4); %I A079226 %S A079226 1,1,2,3,5,6,15,36,108,301,814,2080,5223,12919,32557,83943,222591,600252,1632814, %T A079226 4440240,12043224,32572225,88081208,238722759,649725756,1776546687,4877740703, %U A079226 13432630929,37063472432,102389547753,283124660567,783681212818,2171890574093,6027795895016 %N A079226 Number of Catalan objects fixed by five-fold application of the gatomorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees). %F A079226 a(n) = A079216(n,5) %R A079226 %O A079226 0,3 %K A079226 nonn,new %A A079226 Antti Karttunen (my_firstname.my_surname@iki.fi) Jan 03 2002 %Y A079226 The fifth row of A079216. The leftmost edge of the triangle A079221 and also its row sums shifted by one. Occurs in A073202 as row 9259542121261050623. Cf. A057546, A079223-A079227. %D A079226 %p A079226 A079226 := n -> A079216bi(n,5); %I A079227 %S A079227 1,1,2,5,14,38,111,332,1029,3232,10374,33679,110722,367252,1228558, %T A079227 4138120,14025473,47792389,163643066,562722427,1942548520,6729230281, %U A079227 23385132060,81503084084,284815902739,997741303308,3503112067273 %N A079227 Number of Catalan objects fixed by six-fold application of the gatomorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees). %F A079227 a(n) = A079216(n,6) %R A079227 %O A079227 0,3 %K A079227 nonn,new %A A079227 Antti Karttunen (my_firstname.my_surname@iki.fi) Jan 03 2002 %Y A079227 The sixth row of A079216. The leftmost edge of the triangle A079222 and also its row sums shifted by one. Occurs in A073202 as row 170143779648513184865579715159814351615. Cf. A057546, A079223-A079226. %D A079227 %p A079227 A079227 := n -> A079216bi(n,6); ----------------------------------------------------------------------------- That's all for now! Yours, Antti