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a(n) = [x^n] (2*n)! * Sum_{k=0..2*n} binomial(x+k,k).
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%I #23 Sep 24 2021 11:33:38

%S 1,5,71,1665,54649,2310945,119753843,7353403057,522289211873,

%T 42137920501677,3807384320667135,380929847762489025,

%U 41811136672902061321,4995760464106519955705,645541681316043216096315,89705032647088734873129825,13340173206548155385625683265,2114001534402053456524492822485

%N a(n) = [x^n] (2*n)! * Sum_{k=0..2*n} binomial(x+k,k).

%H Seiichi Manyama, <a href="/A347989/b347989.txt">Table of n, a(n) for n = 0..325</a>

%F a(n) = (2*n)! * Sum_{k=n..2*n} (2*n+1-k) * |Stirling1(k,n)|/k!.

%F a(n) = [x^(2*n)] ((2*n)!/n!) * (-log(1 - x))^n/(1 - x)^2.

%F From _Vaclav Kotesovec_, Sep 23 2021: (Start)

%F a(n) = [x^n] Gamma(2*n + x + 2) / Gamma(x + 2).

%F a(n) ~ c * d^n * (n-1)!, where d = 8*w^2/(2*w-1), w = -LambertW(-1,-exp(-1/2)/2) and c = 1.5967712192197964362930380385801737624829174112909160160618... (End)

%o (PARI) a(n) = (2*n)!*polcoef(sum(k=n, 2*n, binomial(x+k, k)), n);

%o (PARI) a(n) = (2*n)!*sum(k=n, 2*n, (2*n+1-k)*abs(stirling(k, n, 1))/k!);

%Y Cf. A001706, A001707, A001708, A001709, A008275, A143491, A347987.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Sep 23 2021