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

0,2

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

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

Sequence in context: A243249 A075756 A363008 * A144345 A294158 A209306

Adjacent sequences: A217483 A217484 A217485 * A217487 A217488 A217489

Keyword

nonn

Author

Emanuele Munarini, Oct 04 2012

Status

approved