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A123681 a(n) = (1/(n+1)) * Sum_{k=0..n} C(n+k-1,k)*k! = A123680(n)/(n+1).

%I A123681 #18 Nov 27 2017 12:28:37
%S A123681 1,1,3,19,197,2841,52327,1171871,30899529,937529317,32173291931,
%T A123681 1232093935227,52088478142861,2409578607253169,121067200114483407,
%U A123681 6565538372492694871,382234458749760846737,23777755561583494209981
%N A123681 a(n) = (1/(n+1)) * Sum_{k=0..n} C(n+k-1,k)*k! = A123680(n)/(n+1).
%H A123681 G. C. Greubel, <a href="/A123681/b123681.txt">Table of n, a(n) for n = 0..365</a>
%F A123681 a(n) = (1/(n+1)) * Sum_{k=0..n} k! * [x^k] 1/(1-x)^n.
%F A123681 a(n) ~ 2^(2*n - 1/2) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Nov 27 2017
%e A123681 a(n) = (Pochhammer(n, n + 1)*subfactorial(-2*n - 1) + (-1)^n*subfactorial(-n))/(n+1) where subfactorial(n) = exp(-1)*Gamma(n + 1, -1). - _Peter Luschny_, Oct 18 2017
%p A123681 subfactorial := n -> simplify(exp(-1)*GAMMA(n+1,-1)):
%p A123681 a := n -> (pochhammer(n,n+1)*subfactorial(-2*n-1)+(-1)^n*subfactorial(-n))/(n+1):
%p A123681 seq(simplify(evalc(a(n))), n=0..17); # _Peter Luschny_, Oct 18 2017
%t A123681 Table[1/(n+1) Sum[Binomial[n+k-1,k]k!,{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Dec 14 2012 *)
%o A123681 (PARI) a(n)=sum(k=0,n,binomial(n+k-1,k)*k!)/(n+1)
%Y A123681 Cf. A123680.
%K A123681 nonn
%O A123681 0,3
%A A123681 _Paul D. Hanna_, Oct 05 2006
%E A123681 Definition corrected by _Harvey P. Dale_, Dec 14 2012