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%I #32 Sep 08 2022 08:46:17
%S 1,4,23,168,1473,14988,173007,2228544,31636449,490102164,8219695239,
%T 148262469336,2860241078817,58736954622492,1278727896354687,
%U 29406849577341552,712119108949808193,18108134430393657636,482306685868464422391,13425231879291031821576
%N a(n) = n!*LaguerreL(n, -3).
%C For m > 0, n!*LaguerreL(n, -m) ~ exp(2*sqrt(m*n) - n - m/2) * n^(n + 1/4) / (sqrt(2)*m^(1/4)) * (1 + (3+24*m+4*m^2)/(48*sqrt(m*n))).
%H Alois P. Heinz, <a href="/A277382/b277382.txt">Table of n, a(n) for n = 0..438</a>
%H W. Van Assche, <a href="https://doi.org/10.1137/S0036141099359871">Erratum to "Weighted zero distribution for polynomials orthogonal on an infinite interval"</a>, SIAM J. Math. Anal., 32 (2001), 1169-1170.
%H Oskar Perron, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002168839">Über das Verhalten einer ausgearteten hypergeometrischen Reihe bei unbegrenztem Wachstum eines Parameters</a>, Journal für die reine und angewandte Mathematik (1921), vol. 151, p. 63-78.
%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>
%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F E.g.f.: exp(3*x/(1-x))/(1-x).
%F a(n) = Sum_{k=0..n} 3^k*(n-k)!*binomial(n, k)^2.
%F a(n) ~ exp(2*sqrt(3*n)-n-3/2) * n^(n+1/4) / (sqrt(2) * 3^(1/4)) * (1 + 37/(16*sqrt(3*n))).
%F D-finite with recurrence a(n) = 2*(n+1)*a(n-1) - (n-1)^2*a(n-2).
%F Lim n -> infinity a(n)/(n!*BesselI(0, 2*sqrt(3*n))) = exp(-3/2).
%F a(n) = n! * A160613(n)/A160614(n). - _Alois P. Heinz_, Jun 28 2017
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 3^n * x^n / (n!)^2. - _Ilya Gutkovskiy_, Jul 17 2020
%t Table[n!*LaguerreL[n, -3], {n, 0, 20}]
%t CoefficientList[Series[E^(3*x/(1-x))/(1-x), {x, 0, 20}], x] * Range[0, 20]!
%t Table[Sum[Binomial[n, k]^2 * 3^k * (n-k)!, {k,0,n}], {n, 0, 20}]
%o (PARI) for(n=0,30, print1(n!*(sum(k=0,n, binomial(n,k)*(3^k/k!))), ", ")) \\ _G. C. Greubel_, May 09 2018
%o (Magma) [Factorial(n)*((&+[Binomial(n,k)*(3^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // _G. C. Greubel_, May 09 2018
%Y Cf. A002720, A087912, A277372.
%Y Column k=3 of A289192.
%Y Cf. A160613, A160614.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Oct 12 2016
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