A384623 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 A384622.

1, 1, 0, 1, 1, 0, 1, 2, 7, 0, 1, 3, 15, 75, 0, 1, 4, 24, 164, 989, 0, 1, 5, 34, 268, 2177, 14822, 0, 1, 6, 45, 388, 3585, 32672, 242833, 0, 1, 7, 57, 525, 5235, 53922, 534781, 4253818, 0, 1, 8, 70, 680, 7150, 78972, 882304, 9349160, 78573475, 0, 1, 9, 84, 854, 9354, 108251, 1292456, 15399930, 172255669, 1516124048, 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,5*j).

Example

Square array begins:
  1,      1,      1,      1,       1,       1,       1, ...
  0,      1,      2,      3,       4,       5,       6, ...
  0,      7,     15,     24,      34,      45,      57, ...
  0,     75,    164,    268,     388,     525,     680, ...
  0,    989,   2177,   3585,    5235,    7150,    9354, ...
  0,  14822,  32672,  53922,   78972,  108251,  142218, ...
  0, 242833, 534781, 882304, 1292456, 1772920, 2332044, ...

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, 5*j)));

Crossrefs

Columns k=0..1 give A000007, A384622.

Cf. A379599, A384619, A384620, A384621.

Sequence in context: A356470 A260929 A290789 * A381594 A384859 A384860

Adjacent sequences: A384620 A384621 A384622 * A384624 A384625 A384626

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 05 2025

Status

approved