A194689 a(n) = Sum_{k=0..n} binomial(n,k)*w(k)*w(n-k) where w() = A000296().

1, 0, 2, 2, 14, 42, 222, 1066, 6078, 36490, 238046, 1653610, 12214270, 95361866, 784071966, 6764984362, 61066919230, 575200190986, 5640081557598, 57450510336234, 606773139773054, 6633515763375306, 74950634205257630, 873995513192234410, 10504736507220958142, 129983468625156713354

Offset: 0

Author

N. J. A. Sloane, Sep 01 2011

Keywords

• nonn

References

• D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 771, Problem 37).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..560

Crossrefs

Cf. A000296, A001861, A194689, A339014, A339017, A339027.

Programs

Mathematica

Table[Sum[(-1)^(n-k) * Binomial[n,k] * BellB[k,2] * 2^(n-k), {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jun 25 2022 *)

PARI

N=66;  x='x+O('x^N);
Q(k) = if (k>N, 1, 1 + x + x*k - x/(1 - 2*x*(k+1)/Q(k+1) ) );
gf=1/Q(0);  Vec(Ser(gf))
/* Joerg Arndt, Mar 07 2013 */

PARI

my(N=66, x='x+O('x^N)); Vec(serlaplace(exp(2*(exp(x)-1-x)))) \\ Seiichi Manyama, Nov 20 2020

Formula

G.f.: 1/Q(0) where Q(k) = 1 + x + x*k - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013

G.f.: 1/Q(0), where Q(k)= 1 - x*k - 2*x^2*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 06 2013

E.g.f.: exp(2*(exp(x) - 1 - x)). - Ilya Gutkovskiy, Apr 07 2018

a(0) = 1; a(n) = 2 * Sum_{k=1..n-1} binomial(n-1,k) * a(n-1-k). - Seiichi Manyama, Nov 20 2020

a(n) ~ 4 * n^(n-2) * exp(n/LambertW(n/2) - n - 2) / (sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n-2)). - Vaclav Kotesovec, Jun 26 2022