A082545 a(n) = (2*n)! * Sum_{k=0..n} binomial(n,k)/(n+k)!.
1, 3, 21, 229, 3393, 63591, 1442173, 38398641, 1174226049, 40558249963, 1561734494661, 66335687785533, 3081211226192641, 155369391396527439, 8452596370942940973, 493494408990278911561, 30777323181433121541633, 2042075395611656190239571
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..365
Programs
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Magma
[Factorial(n)*Evaluate(LaguerrePolynomial(n, n), -1): n in [0..40]]; // G. C. Greubel, Aug 11 2022
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Maple
a:= n-> simplify(n!*LaguerreL(n$2, -1)): seq(a(n), n=0..20); # Alois P. Heinz, Jul 27 2017
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Mathematica
Table[n!*LaguerreL[n, n, -1], {n,0,17}] (* Jean-François Alcover, Jun 04 2019 *) -
PARI
a(n) = sum(k=0, n, k!*binomial(n, k)*binomial(2*n, k)); \\ Seiichi Manyama, May 01 2021
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PARI
a(n) = n!*pollaguerre(n, n, -1); \\ Seiichi Manyama, May 01 2021
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SageMath
[factorial(n)*gen_laguerre(n, n, -1) for n in (0..40)] # G. C. Greubel, Aug 11 2022
Formula
a(n) = n!*LaguerreL(n, n, -1).
n*a(n) + (n^3-5*n^2-n+2)*a(n-1) - 2*(n+1)*(2*n-3)*(n-1)^2*a(n-2) = 0. - Vladeta Jovovic, Jul 16 2004
E.g.f.: exp((-2*x+1-(1-4*x)^(1/2))/(2*x))/(1-4*x)^(1/2). - Mark van Hoeij, Oct 31 2011
a(n) ~ n^n*2^(2*n+1/2)/exp(n-1). - Vaclav Kotesovec, Sep 27 2012
a(n) = n!*binomial(2*n,n)*hypergeom([-n], [1+n], -1). - Peter Luschny, May 04 2017
a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n+1). - Ilya Gutkovskiy, Nov 21 2017