A006957 Self-convolution of numbers of preferential arrangements.

1, 2, 7, 32, 185, 1310, 11067, 109148, 1234045, 15752858, 224169407, 3518636504, 60381131265, 1124390692886, 22577494959427, 486212633129300, 11177317486573445, 273173247028616594, 7072436847620016327, 193351544314753174736, 5565941751233499986185

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000670, A217388, A217389.

Programs

Maple

f:= proc(n) option remember; `if`(n<=1, 1,
      add(binomial(n, k) *f(n-k), k=1..n))
    end:
a:= n-> add(f(k)*f(n-k), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 02 2009

Mathematica

t[n_] := Sum[StirlingS2[n, k]*k!, {k, 0, n}]; Table[Sum[t[k]*t[n-k], {k, 0, n}], {n, 0, 20}] (* Jean-François Alcover, Apr 09 2014, after Emanuele Munarini *)

Maxima

t(n):=sum(stirling2(n,k)*k!,k,0,n);
makelist(sum(t(k)*t(n-k),k,0,n),n,0,20); /* Emanuele Munarini, Oct 02 2012 */

PARI

a006957(n)=my(x='x+O('x^(n+2))); Vec((x-2*log(2-exp(x)))/(4-exp(x)))[n+1]*(n+1)! \\ Hugo Pfoertner, Dec 27 2024

Formula

a(n) ~ n! / (log(2))^(n+1). - Vaclav Kotesovec, Nov 08 2014

G.f.: (Sum_{j>=0} j!*x^j / Product_{k=1..j} (1 - k*x))^2. - Ilya Gutkovskiy, Apr 06 2019

a(n) = (n+1)! [x^(n+1)] (x-2*log(2-exp(x)))/(4-exp(x)). - Ira M. Gessel, Dec 26 2024

Extensions

More terms from Alois P. Heinz, Feb 02 2009