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%I #19 Aug 27 2022 18:33:50
%S 1,2,9,76,985,17046,366289,9374968,278095761,9375293170,353906211241,
%T 14785127222724,677150215857193,33734100501544366,1816008001717251105,
%U 105048613959883117936,6497985798745934394529,427999600108502895779658
%N a(n) = Sum_{k=0..n} C(n+k-1,k)*k!.
%C a(n) is divisible by (n+1): a(n)/(n+1) = A123681(n).
%H G. C. Greubel, <a href="/A123680/b123680.txt">Table of n, a(n) for n = 0..365</a>
%F a(n) = Pochhammer(n, n + 1)*subfactorial(-2*n - 1) + (-1)^n*subfactorial(-n) where subfactorial(n) = exp(-1)*Gamma(n + 1, -1). - _Peter Luschny_, Oct 18 2017
%F a(n) ~ 2^(2*n - 1/2) * n^n / exp(n). - _Vaclav Kotesovec_, Nov 27 2017
%e Since a(n) = Sum_{k=0..n} k! * [x^k] 1/(1-x)^n, to get a(4),
%e list coefficients of x^0 through x^4 in 1/(1-x)^4, [1,4,10,20,35],
%e then dot product with factorials 0! through 4!, [0!,1!,2!,3!,4! ],
%e so that a(4) = 1*0! + 4*1! + 10*2! + 20*3! + 35*4! = 985.
%p subfactorial := n -> simplify(exp(-1)*GAMMA(n+1,-1)):
%p a := n -> pochhammer(n,n+1)*subfactorial(-2*n-1)+(-1)^n*subfactorial(-n):
%p seq(simplify(evalc(a(n))), n=0..18); # _Peter Luschny_, Oct 18 2017
%t Table[Sum[Binomial[n + k - 1, k]*k!, {k, 0, n}], {n, 0, 20}] (* _G. C. Greubel_, Oct 18 2017 *)
%o (PARI) a(n)=sum(k=0,n,binomial(n+k-1,k)*k!)
%Y Cf. A123681.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Oct 05 2006
%E Definition corrected Oct 27 2006
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