A307064 Expansion of 1 - 1/Sum_{k>=0} k!!*x^k.

0, 1, 1, 0, 3, 1, 18, 13, 155, 168, 1691, 2381, 22022, 37401, 331087, 649036, 5626103, 12372161, 106486594, 257573405, 2220690451, 5824952232, 50593271507, 142387607469, 1250521775454, 3745193283657, 33338037080183, 105558942751948, 953776675614223

Offset: 0

Author

Ilya Gutkovskiy, Mar 21 2019

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..800

Crossrefs

Cf. A000698, A003319, A006882, A292778.

Programs

Magma

m:=80;
F2:= func< n | n mod 2 eq 0 select Round(2^(n/2)*Gamma(n/2+1)) else Round( Gamma((n+3)/2)*Binomial(n+1, Floor((n+1)/2))/2^((n+1)/2) ) >;
R:=PowerSeriesRing(Rationals(), m);
[0] cat Coefficients(R!( 1 - 1/(&+[F2(j)*x^j : j in [0..m+2]]) )); // G. C. Greubel, Jan 24 2024

Mathematica

nmax = 28; CoefficientList[Series[1 - 1/Sum[k!! x^k, {k, 0, nmax}], {x, 0, nmax}], x]
a[0] = 0; a[n_]:= a[n] = n!! - Sum[k!! a[n-k], {k,n-1}];
Table[a[n], {n, 0, 28}]

SageMath

from sympy import factorial2
m=80;
def f(x): return 1 - 1/sum(factorial2(k)*x^k for k in range(m+1))
def A307063_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307063_list(m) # G. C. Greubel, Jan 24 2024

Formula

a(0) = 0; a(n) = n!! - Sum_{k=1..n-1} k!!*a(n-k).