A118930 E.g.f.: A(x) = exp( Sum_{n>=0} x^(2^n)/2^(2^n-1) ).

1, 1, 2, 4, 13, 41, 166, 652, 3494, 18118, 114076, 681176, 5016892, 35377564, 288204008, 2232198256, 21124254181, 191779964597, 2011347229114, 19840403629108, 231266808172181, 2553719667653281, 31743603728993542

Offset

0,3

Comments

Equals invariant column vector V that satisfies matrix product A100861*V = V, where Bessel numbers A100861(n,k) = n!/[k!(n-2k)!*2^k] give the number of k-matchings of the complete graph K(n).

Equals Lim_{n->inf.} A144299^n, if A144299 is considered an infinite lower triangular matrix. - Gary W. Adamson, Dec 08 2008

Links

Table of n, a(n) for n=0..22.

Formula

a(n) = Sum_{k=0..[n/2]} n!/[k!*(n-2*k)!*2^k] * a(k), with a(0)=1. a(n) = Sum_{k=0..[n/2]} A100861(n,k)*a(k), with a(0)=1.

Example

E.g.f. A(x) = exp( x + x^2/2 + x^4/2^3 + x^8/2^7 + x^16/2^15 +...)
= 1 + 1*x + 2*x^2/2! + 4*x^3/3! + 13*x^4/4! + 41*x^5/5!+ 166*x^6/6!+...
Using coefficients A100861(n,k) = n!/[k!(n-2k)!*2^k]:
a(5) = 1*a(0) +10*a(1) +15*a(2) = 1*1 +10*1 +15*2 = 41.
a(6) = 1*a(0) +15*a(1) +45*a(2) +15*a(3) = 1*1 +15*1 +45*2 +15*4 = 166.

Maple

A118930 := proc(n)
    option remember;
    if n<= 1 then
        1 ;
    else
        n!*add(procname(k)/k!/(n-2*k)!/2^k, k=0..n/2) ;
    end if;
end proc;
seq(A118930(n), n=0..10) ; # R. J. Mathar, Aug 19 2014

Mathematica

a[n_] := a[n] = If[n==0, 1, Sum[Binomial[n, 2k] (2k-1)!! a[k], {k, 0, n/2}]];
a /@ Range[0, 22] (* Jean-François Alcover, Mar 26 2020 *)

Prog

(PARI) {a(n)=if(n==0, 1, sum(k=0, n\2, n!/(k!*(n-2*k)!*2^k)*a(k)))}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Defined by E.G.F.: */
{a(n)=n!*polcoeff( exp(sum(k=0, #binary(n), x^(2^k)/2^(2^k-1))+x*O(x^n)), n, x)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A100861; variants: A118932, A118935.

Equals row sums of triangle A152685. - Gary W. Adamson, Dec 10 2008

Cf. A144299. - Gary W. Adamson, Dec 08 2008

Sequence in context: A148256 A163136 A325578 * A355194 A087214 A259239

Adjacent sequences: A118927 A118928 A118929 * A118931 A118932 A118933

Keyword

nonn

Author

Paul D. Hanna, May 06 2006

Status

approved