A049324 A convolution triangle of numbers generalizing Pascal's triangle A007318.

1, 3, 1, 3, 6, 1, 0, 15, 9, 1, 0, 18, 36, 12, 1, 0, 9, 81, 66, 15, 1, 0, 0, 108, 216, 105, 18, 1, 0, 0, 81, 459, 450, 153, 21, 1, 0, 0, 27, 648, 1305, 810, 210, 24, 1, 0, 0, 0, 594, 2673, 2970, 1323, 276, 27, 1, 0, 0, 0, 324, 3915, 7938, 5859, 2016, 351, 30, 1

Offset

1,2

Links

Table of n, a(n) for n=1..66.

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*p(2, x))^m, p(2, x) := 1+3*x+3*x^2 (row polynomial of A033842(2, m)).

Example

Triangle begins:
  {1};
  {3,1};
  {3,6,1};
  {0,15,9,1};
  {0,18,36,12,1};
  ...

Prog

(PARI) a(n, m) = if (n<m, 0, if (m==0, 0, if (n==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, Sep 29 2025

Crossrefs

a(n, m) := s1(-2, n, m), a member of a sequence of triangles including s1(0, n, m)= A023531(n, m) (unit matrix) and s1(2, n, m)=A007318(n-1, m-1) (Pascal's triangle). s1(-1, n, m)= A030528.

Cf. A049348, A049404.

Sequence in context: A146908 A279249 A281893 * A356444 A256095 A181843

Adjacent sequences: A049321 A049322 A049323 * A049325 A049326 A049327

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang

Status

approved