A384751 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. B(x)^k, where B(x) is the e.g.f. of A384749.

1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 12, 73, 0, 1, 4, 21, 176, 1881, 0, 1, 5, 32, 315, 4496, 73281, 0, 1, 6, 45, 496, 8025, 172672, 3919453, 0, 1, 7, 60, 725, 12672, 304803, 9107008, 271474953, 0, 1, 8, 77, 1008, 18665, 477504, 15874605, 622823168, 23404227185, 0

Offset

0,8

Links

Table of n, a(n) for n=0..54.

Formula

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} (2*n-2*j+k)^(j-1) * binomial(n,j) * A(n-j,2*j).

Example

Square array begins:
  1,     1,      1,      1,      1,      1, ...
  0,     1,      2,      3,      4,      5, ...
  0,     5,     12,     21,     32,     45, ...
  0,    73,    176,    315,    496,    725, ...
  0,  1881,   4496,   8025,  12672,  18665, ...
  0, 73281, 172672, 304803, 477504, 699925, ...

Prog

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

Crossrefs

Columns k=0..1 give A000007, A384749.

Sequence in context: A384718 A379168 A384721 * A396973 A396499 A341268

Adjacent sequences: A384748 A384749 A384750 * A384752 A384753 A384754

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 09 2025

Status

approved