This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A277419 #15 Feb 16 2025 08:33:36
%S A277419 1,6,142,5676,318744,23046370,2038090320,213094791840,25714702990720,
%T A277419 3517403388684030,537798502938028800,90890936781714193300,
%U A277419 16825134146527678233600,3385560150770468257273050,735772370353606135149107200,171753027520961356975091493000
%N A277419 a(n) = n!*LaguerreL(n, -5*n).
%H A277419 G. C. Greubel, <a href="/A277419/b277419.txt">Table of n, a(n) for n = 0..250</a>
%H A277419 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A277419 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%F A277419 a(n) = n! * Sum_{k=0..n} binomial(n, k) * 5^k * n^k / k!.
%F A277419 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 A277419 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 A277419 Table[n!*LaguerreL[n, -5*n], {n, 0, 20}]
%t A277419 Flatten[{1, Table[n!*Sum[Binomial[n, k] * 5^k * n^k / k!, {k, 0, n}], {n, 1, 20}]}]
%o A277419 (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 A277419 (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 A277419 Cf. A277373, A277391, A277392, A277418, A277420, A277421, A277422.
%Y A277419 Cf. A002720, A087912, A277382.
%K A277419 nonn
%O A277419 0,2
%A A277419 _Vaclav Kotesovec_, Oct 14 2016