A384978 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 A384975.

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, 16, 0, 1, 4, 15, 40, 77, 0, 1, 5, 22, 73, 202, 303, 0, 1, 6, 30, 116, 387, 888, 718, 0, 1, 7, 39, 170, 645, 1851, 2914, -4934, 0, 1, 8, 49, 236, 990, 3304, 7267, -3544, -108553, 0, 1, 9, 60, 315, 1437, 5376, 14616, 8463, -205605, -1275290, 0

Offset

0,8

Links

Table of n, a(n) for n=0..65.

Formula

Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-5*n+5*j+k-1,j-1) * b(n-j,3*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,   4,    9,   15,    22,    30,    39, ...
  0,  16,   40,   73,   116,   170,   236, ...
  0,  77,  202,  387,   645,   990,  1437, ...
  0, 303,  888, 1851,  3304,  5376,  8214, ...
  0, 718, 2914, 7267, 14616, 25980, 42579, ...

Prog

(PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-5*n+5*j+k-1, j-1)*b(n-j, 3*j)/j));
a(n, k) = b(n, -k);

Crossrefs

Columns k=0..1 give A000007, A384975.

Sequence in context: A173004 A384944 A385019 * A378323 A378290 A118343

Adjacent sequences: A384975 A384976 A384977 * A384979 A384980 A384981

Keyword

sign,tabl

Author

Seiichi Manyama, Jun 14 2025

Status

approved