A384619 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of (B(x)/x)^k, where B(x) is the g.f. of A213591.

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, 24, 0, 1, 4, 15, 56, 178, 0, 1, 5, 22, 97, 420, 1512, 0, 1, 6, 30, 148, 738, 3572, 14152, 0, 1, 7, 39, 210, 1145, 6300, 33328, 142705, 0, 1, 8, 49, 284, 1655, 9832, 58702, 334354, 1528212, 0, 1, 9, 60, 371, 2283, 14321, 91640, 586635, 3559310, 17211564, 0

Offset

0,8

Links

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

Formula

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

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,    24,    56,    97,   148,    210,    284, ...
  0,   178,   420,   738,  1145,   1655,   2283, ...
  0,  1512,  3572,  6300,  9832,  14321,  19938, ...
  0, 14152, 33328, 58702, 91640, 133720, 186753, ...

Prog

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

Crossrefs

Columns k=0..1 give A000007, A213591(n+1).

Cf. A379599, A384620, A384621, A384623.

Sequence in context: A378323 A378290 A118343 * A381602 A394901 A309148

Adjacent sequences: A384616 A384617 A384618 * A384620 A384621 A384622

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 04 2025

Status

approved