A384985 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 A384982.

1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 12, 37, 0, 1, 4, 21, 104, 441, 0, 1, 5, 32, 207, 1328, 6201, 0, 1, 6, 45, 352, 2841, 20512, 106813, 0, 1, 7, 60, 545, 5184, 47403, 381568, 1906941, 0, 1, 8, 77, 792, 8585, 92544, 941805, 7753664, 30468273, 0, 1, 9, 96, 1099, 13296, 162925, 1949824, 20868375, 160665856, 55523377, 0

Offset

0,8

Comments

A(10,1) = -38740562379.

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} (-2*n+j+k)^(j-1) * binomial(n,j) * b(n-j,j). Then A(n,k) = b(n,-k).

Example

Square array begins:
  1,    1,     1,     1,     1,      1, ...
  0,    1,     2,     3,     4,      5, ...
  0,    5,    12,    21,    32,     45, ...
  0,   37,   104,   207,   352,    545, ...
  0,  441,  1328,  2841,  5184,   8585, ...
  0, 6201, 20512, 47403, 92544, 162925, ...

Prog

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

Crossrefs

Columns k=0..1 give A000007, A384982.

Sequence in context: A384811 A385061 A384813 * A396994 A384718 A379168

Adjacent sequences: A384982 A384983 A384984 * A384986 A384987 A384988

Keyword

sign,tabl

Author

Seiichi Manyama, Jun 14 2025

Status

approved