A372829 a(n) = n! * Sum_{k=0..floor(n/2)} k! / (2*k)!.

1, 1, 3, 9, 38, 190, 1146, 8022, 64200, 577800, 5778120, 63559320, 762712560, 9915263280, 138813690960, 2082205364400, 33315285870720, 566359859802240, 10194477476803200, 193695072059260800, 3873901441188844800, 81351930264965740800, 1789742465829286214400

Offset: 0

Author

Ilya Gutkovskiy, May 14 2024

Keywords

• nonn

Crossrefs

Cf. A001813, A009179, A084261.

Programs

Maple

A372829 := proc(n)
    add( k!/(2*k)!,k=0..floor(n/2)) ;
    %*n! ;
end proc:
seq(A372829(n),n=0..70) ; # R. J. Mathar, Sep 27 2024

Mathematica

Table[n! Sum[k!/(2 k)!, {k, 0, Floor[n/2]}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(1 + Sqrt[Pi] x Exp[x^2/4] Erf[x/2]/2)/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

PARI

a(n) = n! * sum(k=0, n\2,  k! / (2*k)!); \\ Michel Marcus, May 14 2024

Formula

E.g.f.: (1 + sqrt(Pi) * x * exp(x^2/4) * erf(x/2) / 2) / (1 - x).

a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * k! * (n-2*k)!.

a(n) ~ n! * (1 + exp(1/4)*sqrt(Pi)*erf(1/2)/2). - Vaclav Kotesovec, May 14 2024

D-finite with recurrence 2*a(n) -2*n*a(n-1) -n*a(n-2) +n*(n-2)*a(n-3)=0. - R. J. Mathar, Sep 27 2024