A158835 Triangle, read by rows, that transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).
1, 1, 1, 4, 2, 1, 27, 11, 3, 1, 254, 94, 21, 4, 1, 3062, 1072, 217, 34, 5, 1, 45052, 15212, 2904, 412, 50, 6, 1, 783151, 257777, 47337, 6325, 695, 69, 7, 1, 15712342, 5074738, 906557, 116372, 12035, 1082, 91, 8, 1, 357459042, 113775490, 19910808, 2483706
Offset: 1
Examples
Triangle T begins: 1; 1,1; 4,2,1; 27,11,3,1; 254,94,21,4,1; 3062,1072,217,34,5,1; 45052,15212,2904,412,50,6,1; 783151,257777,47337,6325,695,69,7,1; 15712342,5074738,906557,116372,12035,1082,91,8,1; 357459042,113775490,19910808,2483706,246596,20859,1589,116,9,1; 9094926988,2861365660,492818850,60168736,5801510,470928,33747,2232,144,10,1; ... Array A158825 of coefficients in iterations of x*C(x) begins: 1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,...; 1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...; 1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,...; 1,4,20,110,640,3870,24084,153306,993978,6544242,43652340,...; 1,5,30,195,1330,9380,67844,500619,3755156,28558484,...; 1,6,42,315,2464,19852,163576,1372196,11682348,100707972,...; 1,7,56,476,4200,38052,351792,3305484,31478628,303208212,...; 1,8,72,684,6720,67620,693048,7209036,75915708,807845676,...; 1,9,90,945,10230,113190,1273668,14528217,167607066,...; 1,10,110,1265,14960,180510,2212188,27454218,344320262,...; ... This triangle transforms diagonals of A158825 into each other: T*A158831 = A158832; T*A158832 = A158833; T*A158833 = A158834; where: A158831 = [1,1,6,54,640,9380,163576,3305484,...]; A158832 = [1,2,12,110,1330,19852,351792,7209036,...]; A158833 = [1,3,20,195,2464,38052,693048,14528217,...]; A158834 = [1,4,30,315,4200,67620,1273668,27454218,...].
Links
- Paul D. Hanna, Table of n, a(n), n = 1..496 (rows 1..31).
Crossrefs
Programs
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PARI
{T(n, k)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+2))), M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]} for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
Extensions
Edited by N. J. A. Sloane, Oct 04 2010, to make entries, offset, b-file and link to b-file all consistent.
Comments