A384976 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 A384951.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 3, 0, 1, 4, 9, 10, 5, 0, 1, 5, 14, 22, 20, 6, 0, 1, 6, 20, 40, 51, 34, 2, 0, 1, 7, 27, 65, 105, 105, 45, -20, 0, 1, 8, 35, 98, 190, 248, 188, 18, -102, 0, 1, 9, 44, 140, 315, 501, 526, 255, -175, -312, 0, 1, 10, 54, 192, 490, 912, 1200, 956, 63, -836, -795, 0

Offset

0,8

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(-n+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, 2,  5,   9,  14,   20,   27, ...
  0, 3, 10,  22,  40,   65,   98, ...
  0, 5, 20,  51, 105,  190,  315, ...
  0, 6, 34, 105, 248,  501,  912, ...
  0, 2, 45, 188, 526, 1200, 2408, ...

Prog

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

Crossrefs

Columns k=0..1 give A000007, A384951.

Sequence in context: A384899 A384900 A296068 * A144064 A172236 A191646

Adjacent sequences: A384973 A384974 A384975 * A384977 A384978 A384979

Keyword

sign,tabl

Author

Seiichi Manyama, Jun 14 2025

Status

approved