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a(n) = n! * [x^n] exp((n + 1)*x + x^2/2).
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%I #14 Apr 08 2018 08:20:20

%S 1,2,10,76,778,10026,155884,2839880,59339004,1399069450,36746349496,

%T 1064024248068,33676500286840,1156685567791586,42850609041047760,

%U 1703182952266379536,72299420602524921616,3264579136056004359570,156238968782480840396704,7900247992586138688381500

%N a(n) = n! * [x^n] exp((n + 1)*x + x^2/2).

%H Muniru A Asiru, <a href="/A301741/b301741.txt">Table of n, a(n) for n = 0..200</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F a(n) = [x^n] 1/(1 - (n + 1)*x - x^2/(1 - (n + 1)*x - 2*x^2/(1 - (n + 1)*x - 3*x^2/(1 - ...)))), a continued fraction.

%F a(n) = Sum_{k=0..floor(n/2)} n!*(n + 1)^(n-2*k)/(2^k*k!*(n - 2*k)!).

%F a(n) ~ exp(3/2) * n^n. - _Vaclav Kotesovec_, Apr 08 2018

%t Table[n! SeriesCoefficient[Exp[(n + 1) x + x^2/2], {x, 0, n}], {n, 0, 19}]

%t Table[SeriesCoefficient[1/(1 - (n + 1) x + ContinuedFractionK[-k x^2, 1 - (n + 1) x, {k, 1, n}]), {x, 0, n}], {n, 0, 19}]

%t Table[Sum[n! (n + 1)^(n - 2 k)/(2^k k! (n - 2 k)!), {k, 0, Floor[n/2]}], {n, 0, 19}]

%o (GAP) List([0..10],n->Sum([0..Int(n/2)],k->Factorial(n)*(n+1)^(n-2*k)/(2^k*Factorial(k)*Factorial(n-2*k)))); # _Muniru A Asiru_, Mar 26 2018

%Y Cf. A000085, A005425, A202834, A202879.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Mar 26 2018