A217486 Binomial convolution of the numbers in sequence A080253.

1, 6, 52, 600, 8656, 149856, 3026752, 69866880, 1814338816, 52350752256, 1661575754752, 57531530434560, 2158011794968576, 87173881613869056, 3772959800981143552, 174183372619165040640, 8543978588021450407936, 443748799382401230176256

Offset: 0

Author

Emanuele Munarini, Oct 04 2012

Keywords

• nonn

Links

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

Crossrefs

Cf. A080253, A000670, A217483, A217484, A217485, A217487, A217488.

Programs

Mathematica

t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[Sum[Binomial[n,k]c[k]c[n-k], {k,0,n}], {n,0,100}]; Table[2^n t[n+1], {n,0,100}]
With[{nn=20},CoefficientList[Series[Exp[2x]/(2-Exp[2x])^2,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Oct 09 2017 *)

Maxima

t(n):=sum(stirling2(n,k)*k!,k,0,n);
c(n):=sum(binomial(n,k)*2^k*t(k),k,0,n);
makelist(sum(binomial(n,k)*c(k)*c(n-k),k,0,n),n,0,10);
makelist(2^n*t(n+1),n,0,40);

Sage

def A217486(n):
    return 2^n*add(add((-1)^(j-i)*binomial(j,i)*i^(n+1) for i in range(n+2)) for j in range(n+2))
[A217486(n) for n in range(18)] # Peter Luschny, Jul 22 2014

Formula

a(n) = sum(binomial(n,k)*c(k)*c(n.k),k=0..n), where c(n) = A080253(n).

a(n) = 2^n*t(n+1), where t(n) = ordered Bell numbers (A000670).

E.g.f. exp(2*x)/(2-exp(2*x))^2.

G.f.: 1/G(0) where G(k) = 1 - x*3*(2*k+2) + x^2*(k+1)*(k+2)*(1-3^2)/G(k+1) ; (continued fraction due to T. J. Stieltjes). - Sergei N. Gladkovskii, Jan 11 2013.

a(n) ~ n!*n*2^(n-1)/(log(2))^(n+2). - Vaclav Kotesovec, Aug 11 2013