A384987 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 A384984.

1, 1, 0, 1, 1, 0, 1, 2, 9, 0, 1, 3, 20, 133, 0, 1, 4, 33, 320, 3185, 0, 1, 5, 48, 567, 7920, 88521, 0, 1, 6, 65, 880, 14529, 232832, 2709625, 0, 1, 7, 84, 1265, 23360, 448203, 7695232, 59590189, 0, 1, 8, 105, 1728, 34785, 752064, 15740001, 220228416, -2800437663, 0

Offset

0,8

Links

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

Formula

Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} (-6*n+5*j+k)^(j-1) * binomial(n,j) * b(n-j,3*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,     9,     20,     33,     48,      65, ...
  0,   133,    320,    567,    880,    1265, ...
  0,  3185,   7920,  14529,  23360,   34785, ...
  0, 88521, 232832, 448203, 752064, 1164125, ...

Prog

(PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-6*n+5*j+k)^(j-1)*binomial(n, j)*b(n-j, 3*j)));
a(n, k) = b(n, -k);

Crossrefs

Columns k=0..1 give A000007, A384984.

Sequence in context: A324330 A197294 A385063 * A396588 A396502 A395148

Adjacent sequences: A384984 A384985 A384986 * A384988 A384989 A384990

Keyword

sign,tabl

Author

Seiichi Manyama, Jun 14 2025

Status

approved