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

Author

Paul D. Hanna, Mar 28 2009, Mar 29 2009

Keywords

• nonn
• tabl

Comments

Conjecture: n-th reversed row polynomial is t_n where we start with vector v of fixed length m with elements v_i = 1, then set t := v and for i=1..m-1, for j=1..i, for k=j+1..i+1 apply v_k := v_k + z*v_{k-1} and t_{i+1} := v_{i+1} (after ending each cycle for j). - Mikhail Kurkov, Sep 03 2024

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

Cf. columns: A158836, A158837, A158838, A158839, row sums: A158840.

Cf. A158825, A158831, A158832, A158833, A158834, variant: A135080.

Programs

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.