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
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 06 2006
STATUS
approved