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a(n) = n! * Sum_{k=0..floor(n/2)} k! / (2*k)!.
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%I #15 Sep 27 2024 05:41:43

%S 1,1,3,9,38,190,1146,8022,64200,577800,5778120,63559320,762712560,

%T 9915263280,138813690960,2082205364400,33315285870720,566359859802240,

%U 10194477476803200,193695072059260800,3873901441188844800,81351930264965740800,1789742465829286214400

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

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

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

%F a(n) ~ n! * (1 + exp(1/4)*sqrt(Pi)*erf(1/2)/2). - _Vaclav Kotesovec_, May 14 2024

%F 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

%p A372829 := proc(n)

%p add( k!/(2*k)!,k=0..floor(n/2)) ;

%p %*n! ;

%p end proc:

%p seq(A372829(n),n=0..70) ; # _R. J. Mathar_, Sep 27 2024

%t Table[n! Sum[k!/(2 k)!, {k, 0, Floor[n/2]}], {n, 0, 22}]

%t 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]!

%o (PARI) a(n) = n! * sum(k=0, n\2, k! / (2*k)!); \\ _Michel Marcus_, May 14 2024

%Y Cf. A001813, A009179, A084261.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 14 2024