A158838 -id:A158838 - cp's OEIS frontend

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

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

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

			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,...].

Paul D. Hanna, Table of n, a(n), n = 1..496 (rows 1..31).

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

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

{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(""))

Edited by N. J. A. Sloane, Oct 04 2010, to make entries, offset, b-file and link to b-file all consistent.

A158836 Column 0 of triangle A158835.

Original entry on oeis.org

1, 1, 4, 27, 254, 3062, 45052, 783151, 15712342, 357459042, 9094926988, 255939571048, 7893741230500, 264806871279676, 9600056691219936, 374033821840909263, 15586672520501193866, 691789220336675178652

Offset: 0

Paul D. Hanna, Mar 28 2009

nonn

Triangle A158835 transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

Paul D. Hanna, Table of n, a(n), n = 0..50.

Cf. A158835, A158837, A158838, A158839, A158840.

{a(n)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+2))),M,N,P);M=matrix(n+2, n+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(n+1, n+1, r, c, M[r, c]); P=matrix(n+1, n+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1,1]}

A158837 Column 1 of triangle A158835.

Original entry on oeis.org

1, 2, 11, 94, 1072, 15212, 257777, 5074738, 113775490, 2861365660, 79763482974, 2440866020252, 81343355108428, 2932370770780016, 113695507437209845, 4717853729131352186, 208615291319607614600, 9792578421235713418464

Offset: 0

Paul D. Hanna, Mar 28 2009

nonn

Triangle A158835 transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

Cf. A158835, A158836, A158838, A158839, A158840.

{a(n)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+3))),M,N,P);M=matrix(n+3, n+3, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(n+2, n+2, r, c, M[r, c]); P=matrix(n+2, n+2, r, c, M[r+1, c]); (P~*N~^-1)[n+2,2]}

A158840 Row sums of triangle A158835.

Original entry on oeis.org

1, 2, 7, 42, 374, 4391, 63637, 1095362, 21823226, 493898216, 12515588806, 351062669154, 10798972965266, 361471373319171, 13080119556342713, 508813238759275712, 21174032937728251318, 938646693399848483498

Offset: 0

Paul D. Hanna, Mar 28 2009

nonn

Triangle A158835 transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

Cf. A158835, A158836, A158837, A158838, A158839.

{a(n)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+2))), M, N, P); M=matrix(n+2, n+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(n+1, n+1, r, c, M[r, c]); P=matrix(n+1, n+1, r, c, M[r+1, c]); sum(k=0,n,(P~*N~^-1)[n+1, k+1])}

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A158836 Column 0 of triangle A158835.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A158837 Column 1 of triangle A158835.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A158840 Row sums of triangle A158835.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI