Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #14 Sep 08 2022 08:46:17
%S 1,6,142,5676,318744,23046370,2038090320,213094791840,25714702990720,
%T 3517403388684030,537798502938028800,90890936781714193300,
%U 16825134146527678233600,3385560150770468257273050,735772370353606135149107200,171753027520961356975091493000
%N a(n) = n!*LaguerreL(n, -5*n).
%H G. C. Greubel, <a href="/A277419/b277419.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) * 5^k * n^k / k!.
%F a(n) ~ sqrt(1/2 + 7/(6*sqrt(5))) * ((7 + 3*sqrt(5))/2)^n * exp((-7 + 3*sqrt(5))*n/2) * n^n.
%F Equivalently, a(n) ~ phi^(4*n + 2) * n^n / (sqrt(3) * 5^(1/4) * exp(n/phi^4)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 06 2021
%t Table[n!*LaguerreL[n, -5*n], {n, 0, 20}]
%t Flatten[{1, Table[n!*Sum[Binomial[n, k] * 5^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)*5^k*n^k/k!), ", ")) \\ _G. C. Greubel_, May 15 2018
%o (Magma) [Factorial(n)*(&+[Binomial(n,k)*5^k*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, May 15 2018
%Y Cf. A277373, A277391, A277392, A277418, A277420, A277421, A277422.
%Y Cf. A002720, A087912, A277382.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Oct 14 2016