A384864 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A213091.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 4, 0, 1, 5, 10, 13, 13, 11, 0, 1, 6, 15, 24, 30, 34, 31, 0, 1, 7, 21, 40, 59, 78, 96, 98, 0, 1, 8, 28, 62, 105, 156, 220, 296, 317, 0, 1, 9, 36, 91, 174, 286, 442, 669, 952, 1070, 0, 1, 10, 45, 128, 273, 492, 820, 1336, 2136, 3182, 3685, 0

Offset

0,8

Comments

A(80,1) = -862046378563961332360903978002479001059082.

Links

Table of n, a(n) for n=0..77.

Formula

Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-2*n+3*j+k-1,j-1) * b(n-j,j)/j. Then A(n,k) = b(n,-k).

Example

Square array begins:
  1,  1,  1,   1,   1,   1,    1, ...
  0,  1,  2,   3,   4,   5,    6, ...
  0,  1,  3,   6,  10,  15,   21, ...
  0,  2,  6,  13,  24,  40,   62, ...
  0,  4, 13,  30,  59, 105,  174, ...
  0, 11, 34,  78, 156, 286,  492, ...
  0, 31, 96, 220, 442, 820, 1437, ...

Prog

(PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-2*n+3*j+k-1, j-1)*b(n-j, j)/j));
a(n, k) = b(n, -k);

Crossrefs

Columns k=0..1 give A000007, A213091.

Sequence in context: A286335 A291652 A378320 * A381566 A071569 A378321

Adjacent sequences: A384861 A384862 A384863 * A384865 A384866 A384867

Keyword

sign,tabl

Author

Seiichi Manyama, Jun 11 2025

Status

approved