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a(n) = n!*LaguerreL(n, -2*n).
11

%I #16 Sep 08 2022 08:46:17

%S 1,3,34,654,17688,616120,26252496,1322624016,76909665664,

%T 5069558461824,373529452588800,30422117430022912,2713911389090970624,

%U 263171888496899625984,27563036166079327578112,3100736138961250867968000,372888702864658105915244544

%N a(n) = n!*LaguerreL(n, -2*n).

%H G. C. Greubel, <a href="/A277391/b277391.txt">Table of n, a(n) for n = 0..250</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>

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

%F a(n) ~ (1 + sqrt(3))^(2*n+1) * n^n / (3^(1/4) * 2^(n+1) * exp((2 - sqrt(3))*n)).

%t Table[n!*LaguerreL[n, -2*n], {n, 0, 20}]

%t Flatten[{1, Table[n!*Sum[Binomial[n, k]*2^k*n^k/k!, {k, 0, n}], {n, 1, 20}]}]

%o (PARI) for(n=0, 30, print1(n!*sum(k=0, n, binomial(n,k)*2^k*n^k/k!), ", ")) \\ _G. C. Greubel_, May 15 2018

%o (Magma) [Factorial(n)*(&+[Binomial(n,k)*2^k*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, May 15 2018

%Y Cf. A002720, A087912, A277382.

%Y Cf. A277373, A277392, A277418, A277419, A277420, A277421, A277422.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Oct 12 2016