OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Emanuele Munarini, Oct 04 2012
STATUS
approved