A295385 a(n) = n!*Sum_{k=0..n} binomial(2*n,n-k)*n^k/k!.
1, 3, 32, 579, 14736, 483115, 19376928, 918980139, 50306339072, 3121729082739, 216541483852800, 16603614676249843, 1394473165806440448, 127308860552307549531, 12553171419275174137856, 1329537514269062031406875, 150531055969843353812533248, 18143286205523964035258551651
Offset: 0
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..335
- Eric Weisstein's World of Mathematics, Laguerre Polynomial
- Wikipedia, Laguerre polynomials
- Index entries for sequences related to Laguerre polynomials
Programs
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Magma
[Factorial(n)*(&+[Binomial(2*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018
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Mathematica
Table[n! SeriesCoefficient[Exp[n x/(1 - x)]/(1 - x)^(n + 1), {x, 0, n}], {n, 0, 17}] Table[n! LaguerreL[n, n, -n], {n, 0, 17}] Table[(-1)^n HypergeometricU[-n, n + 1, -n], {n, 0, 17}] Join[{1}, Table[n! Sum[Binomial[2 n, n - k] n^k/k!, {k, 0, n}], {n, 1, 17}]] -
PARI
for(n=0,30, print1(n!*sum(k=0,n, binomial(2*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018
Formula
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(n+1).
a(n) = n!*Laguerre(n,n,-n).
a(n) ~ 2^(n - 1/2) * (1 + sqrt(2))^(n + 1/2) * n^n / exp((2 - sqrt(2))*n). - Vaclav Kotesovec, Nov 21 2017