A049326 A convolution triangle of numbers generalizing Pascal's triangle A007318.

1, 10, 1, 50, 20, 1, 125, 200, 30, 1, 125, 1250, 450, 40, 1, 0, 5250, 4375, 800, 50, 1, 0, 15000, 30375, 10500, 1250, 60, 1, 0, 28125, 157500, 100500, 20625, 1800, 70, 1, 0, 31250, 621875, 740000, 250625, 35750, 2450, 80, 1, 0, 15625, 1875000, 4318750, 2375000, 525750, 56875, 3200, 90, 1

Offset

1,2

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) = 5*(5*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(4, x))^m, p(4, x) := 1+10*x+50*x^2+125*x^3+125*x^4 (row polynomial of A033842(4, m)).

Example

Triangle begins:
  {1};
  {10,1};
  {50,20,1};
  {125,200,30,1};
  {125,1250,450,40,1};
  ...

Prog

(PARI) a(n, m) = if (n<m, 0, if (m==0, 0, if ((n==1) && (m==1), 1, 5*(5*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

a(n, m) := s1(-4, 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. A049350.

Sequence in context: A115097 A050313 A116574 * A146537 A050304 A143471

Adjacent sequences: A049323 A049324 A049325 * A049327 A049328 A049329

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang

Extensions

More terms from Michel Marcus, Oct 02 2025

Status

approved