A124373 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k*(k+1)/2*x).

1, 1, 2, 6, 25, 135, 909, 7417, 71698, 806968, 10427825, 152915697, 2519879761, 46276398129, 940296067422, 21007099850230, 513172107841525, 13640345170943527, 392780078386164389, 12204609567437300313, 407757149671568266678, 14600807659376773500696

Offset: 0

Author

Paul D. Hanna, Oct 28 2006

Keywords

• nonn

Comments

Starting (1, 2, 6, 25, ...) = row sums of triangle A080248. - Gary W. Adamson, Jul 11 2011

Examples

			Also generated by iterated binomial transforms in the following way:
  [1,2,6,25,135,909,7417,71698,...]  = binomial([1,1,3,12,64,433,3567,...]);
  [1,3,12,64,433,3567,34905,...]     = binomial^2([1,1,4,20,129,1045,...]);
  [1,4,20,129,1045,10209,117069,...] = binomial^3([1,1,5,30,226,2121,...]);
  [1,5,30,226,2121,23919,314605,...] = binomial^4([1,1,6,42,361,3835,...]);
  [1,6,42,361,3835,48885,724569,...] = binomial^5([1,1,7,56,540,6385,...]);
  [1,7,56,540,6385,90519,1490457,..] = binomial^6([1,1,8,72,769,9993,...]);
etc.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..358

Crossrefs

Cf. A000110, A000217, A080248.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m)*m*(m+1)/2 +b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 10 2019

Mathematica

b[n_, m_] := b[n, m] = If[n == 0, 1, b[n-1, m] m(m+1)/2 + b[n-1, m+1]];
a[n_] := b[n, 0];
a /@ Range[0, 23] (* Jean-François Alcover, Nov 02 2020, after Alois P. Heinz *)

PARI

a(n)=polcoeff(sum(k=0,n,x^k/prod(j=0,k,1-j*(j+1)/2*x+x*O(x^n))),n)

Formula

O.g.f.: A(x) = 1 + x/(1-x) + x^2/((1-x)*(1-3x)) + x^3/((1-x)*(1-3x)*(1-6x)) + x^3/((1-x)*(1-3x)*(1-6x)*(1-10x)) + ...

G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-(k+1)*(k+2)*x/2)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2013