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%I #18 Sep 08 2022 08:46:17
%S 1,9,322,19446,1649688,180184120,24070390992,3801662863152,
%T 692979602529664,143184960501077376,33069665092749868800,
%U 8442378658666161822976,2360674573114695421197312,717531421372546588398529536,235551703250624390582942574592
%N a(n) = n!*LaguerreL(n, -8*n).
%C In general, if m > 0 and a(n) = n!*LaguerreL(n, -m*n), then a(n) ~ sqrt(1/2 + (m+2)/(2*sqrt(m*(m+4)))) * (2+m+sqrt(m*(m+4)))^n * exp(n*(sqrt(m*(m+4))-m-2)/2) * n^n / 2^n.
%C For m > 4, (-1)^n * n! * LaguerreL(n, m*n) ~ sqrt(1/2 + (m-2)/(2*sqrt(m*(m-4)))) * exp((m - 2 - sqrt(m*(m-4)))*n/2) * ((m - 2 + sqrt(m*(m-4)))/2)^n * n^n. - _Vaclav Kotesovec_, Feb 20 2020
%H G. C. Greubel, <a href="/A277422/b277422.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) * 8^k * n^k / k!.
%F a(n) ~ sqrt(2 + 5/sqrt(6)) * (5 + 2*sqrt(6))^n * exp((-5 + 2*sqrt(6))*n) * n^n / 2.
%t Table[n!*LaguerreL[n, -8*n], {n, 0, 20}]
%t Flatten[{1, Table[n!*Sum[Binomial[n, k] * 8^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)*(8)^k*n^k/k!), ", ")) \\ _G. C. Greubel_, May 16 2018
%o (Magma) [Factorial(n)*(&+[Binomial(n,k)*(8)^k*n^k/Factorial(k): k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, May 16 2018
%Y Cf. A277373 (m=1), A277391 (m=2), A277392 (m=3), A277418 (m=4), A277419 (m=5), A277420 (m=6), A277421 (m=7).
%Y Cf. A002720, A087912, A277382.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Oct 14 2016
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