A065143 a(n) = Sum_{k=0..n} Stirling2(n,k)*(1+(-1)^k)*2^k/2.

1, 0, 4, 12, 44, 220, 1228, 7196, 45004, 303900, 2201676, 16920860, 136966860, 1163989788, 10364408140, 96463232284, 935872773068, 9440653262620, 98809201693260, 1071131795708188, 12007932126074060

Offset: 0

Author

Karol A. Penson, Oct 17 2001

Keywords

• nonn

Comments

Stirling transform of A199572 (aerated powers of 4).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function
• Eric Weisstein's MathWorld, Bell Polynomial.

Crossrefs

Column k=4 of A357681.

Cf. A199572, A264036, A357598.

Programs

Mathematica

Table[Sum[StirlingS2[n,k]*(1+(-1)^k)*2^k/2,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 06 2014 *)
Table[(BellB[n, 2] + BellB[n, -2])/2, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)

PARI

a(n) = sum(k=0, n, stirling(n,k,2)*(1+(-1)^k)*2^k/2); \\ Michel Marcus, Nov 02 2015

PARI

x='x+O('x^50); Vec(serlaplace(cosh(2*exp(x)-2))) \\ G. C. Greubel, Nov 16 2017

Formula

Representation as a sum of an infinite series: a(n) = exp(2)*Sum_{k = 0..infinity} ((2*k)^n*2^(2*k)/(2*k)!) - sinh(2)*sum_{k = 0..infinity}(k^n*2^k/k!), for n >= 0.

E.g.f.: cosh(2*exp(x)-2). - Vladeta Jovovic, Sep 14 2003

From Vaclav Kotesovec, Aug 06 2014: (Start)

a(n) ~ n^n * cosh(2*exp(r)-2) / (r^n * (exp(n) * sqrt(4*exp(2*r)*r^2/n + 1-n+r))), where r is the root of the equation -2*exp(r)*r*tanh(2-2*exp(r)) = n.

(a(n)/n!)^(1/n) ~ exp(1/LambertW(n/2)) / LambertW(n/2).

(End)

a(n) = (Bell_n(2) + Bell_n(-2))/2, where Bell_n(x) is n-th Bell polynomial. - Vladimir Reshetnikov, Nov 01 2015

a(n) = 1; a(n) = 4 * Sum_{k=0..n-1} binomial(n-1, k) * A357598(k). - Seiichi Manyama, Oct 12 2022