|
%I #19 Mar 26 2020 05:05:44
%S 1,1,2,4,13,41,166,652,3494,18118,114076,681176,5016892,35377564,
%T 288204008,2232198256,21124254181,191779964597,2011347229114,
%U 19840403629108,231266808172181,2553719667653281,31743603728993542
%N E.g.f.: A(x) = exp( Sum_{n>=0} x^(2^n)/2^(2^n-1) ).
%C 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).
%C Equals Lim_{n->inf.} A144299^n, if A144299 is considered an infinite lower triangular matrix. - _Gary W. Adamson_, Dec 08 2008
%F 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.
%e E.g.f. A(x) = exp( x + x^2/2 + x^4/2^3 + x^8/2^7 + x^16/2^15 +...)
%e = 1 + 1*x + 2*x^2/2! + 4*x^3/3! + 13*x^4/4! + 41*x^5/5!+ 166*x^6/6!+...
%e Using coefficients A100861(n,k) = n!/[k!(n-2k)!*2^k]:
%e a(5) = 1*a(0) +10*a(1) +15*a(2) = 1*1 +10*1 +15*2 = 41.
%e a(6) = 1*a(0) +15*a(1) +45*a(2) +15*a(3) = 1*1 +15*1 +45*2 +15*4 = 166.
%p A118930 := proc(n)
%p option remember;
%p if n<= 1 then
%p 1 ;
%p else
%p n!*add(procname(k)/k!/(n-2*k)!/2^k,k=0..n/2) ;
%p end if;
%p end proc;
%p seq(A118930(n),n=0..10) ; # _R. J. Mathar_, Aug 19 2014
%t a[n_] := a[n] = If[n==0, 1, Sum[Binomial[n, 2k] (2k-1)!! a[k], {k, 0, n/2}]];
%t a /@ Range[0, 22] (* _Jean-François Alcover_, Mar 26 2020 *)
%o (PARI) {a(n)=if(n==0,1,sum(k=0,n\2,n!/(k!*(n-2*k)!*2^k)*a(k)))}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) /* Defined by E.G.F.: */
%o {a(n)=n!*polcoeff( exp(sum(k=0,#binary(n),x^(2^k)/2^(2^k-1))+x*O(x^n)),n,x)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A100861; variants: A118932, A118935.
%Y Equals row sums of triangle A152685. - _Gary W. Adamson_, Dec 10 2008
%Y Cf. A144299. - _Gary W. Adamson_, Dec 08 2008
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 06 2006
| 