A030524 A convolution triangle of numbers obtained from A036068.

1, 6, 1, 30, 12, 1, 135, 96, 18, 1, 567, 630, 198, 24, 1, 2268, 3654, 1701, 336, 30, 1, 8748, 19440, 12501, 3564, 510, 36, 1, 32805, 96957, 82296, 31644, 6435, 720, 42, 1, 120285, 459756, 498663, 247536, 66915, 10530, 966, 48, 1, 433026, 2092959, 2830707, 1758942, 605556, 125442, 16065, 1248, 54, 1

Offset

1,2

Comments

a(n,m) := s1p(4; n,m), a member of a sequence of unsigned triangles including s1p(2; n,m)= A007318(n-1,m-1) (Pascal's triangle). Signed version: (-1)^(n-m)*a(n,m) := s1(4; n,m).

Links

Table of n, a(n) for n=1..55.

Wolfdieter Lang, On generalizations of the Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Formula

a(n, m) = 3*(3*m+n-1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0; a(1, 1)=1.

G.f. for m-th column: (x*(1-3*x+3*x^2)/(1-3*x)^3)^m.

Example

Triangle begins:
  {1};
  {6,1};
  {30,12,1};
  {135,96,18,1};
  {567,630,198,24,1};
  ...

Prog

(PARI) a(n, m) = if (n<m, 0, if (m==0, 0, if ((n==1) && (m==1), 1, 3*(3*m+n-1)*a(n-1, m)/n + m*a(n-1, m-1)/n)));
row(n) = vector(n, m, a(n, m)); \\ Michel Marcus, Oct 02 2025

Crossrefs

Cf. A030523, A043553. a(n, 1)= A036068(n-1). Row sums = A043553(n).

Sequence in context: A120105 A120101 A178726 * A327022 A241171 A051930

Adjacent sequences: A030521 A030522 A030523 * A030525 A030526 A030527

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang

Extensions

More terms from Michel Marcus, Oct 02 2025

Status

approved