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%I #64 Aug 21 2022 04:43:08
%S 1,4,31,358,5509,106096,2456299,66471826,2059640713,71920704124,
%T 2794938616471,119653108240414,5595650767265101,283841520215780008,
%U 15523069639558351459,910529206043204428426,57023540590242398853649,3797750659849704886903156,268025698704886063968108943
%N a(n) = Sum_{k=0..n} k! * binomial(n,k) * binomial(2*n+1,k).
%C Let A(x) be the e.g.f. of this sequence, and B(x) be the e.g.f. of A082545, then A(x)/B(x) = C(x) where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108). This follows from the fact that this sequence and A082545 form adjacent semi-diagonals of table A088699. - _Paul D. Hanna_, Aug 16 2022
%H Seiichi Manyama, <a href="/A343832/b343832.txt">Table of n, a(n) for n = 0..365</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 a(n) = (2*n+1)! * Sum_{k=0..n} binomial(n,k)/(k+n+1)!.
%F a(n) = n! * Sum_{k=0..n} binomial(2*n+1,k)/(n-k)!.
%F a(n) = n! * LaguerreL(n, n+1, -1).
%F a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n+2).
%F a(n) == 1 (mod 3).
%F a(n) ~ 2^(2*n + 3/2) * n^n / exp(n-1). - _Vaclav Kotesovec_, May 02 2021
%F From _Paul D. Hanna_, Aug 16 2022: (Start)
%F E.g.f.: exp( (1-2*x - sqrt(1-4*x))/(2*x) ) * (1 - sqrt(1-4*x)) / (2*x*sqrt(1-4*x)), derived from the e.g.f for A082545 given by Mark van Hoeij.
%F E.g.f.: exp(C(x) - 1) * C(x) / sqrt(1-4*x), where C(x) = (1 - sqrt(1-4*x))/(2*x) is the Catalan function (A000108). (End)
%p a := n -> add(k!*binomial(n, k)*binomial(2*n+1, k), k=0..n):
%p a := n -> n!*add(binomial(2*n+1, k)/(n-k)!, k=0..n):
%p a := n -> (-1)^n*KummerU(-n, n+2, -1):
%p a := n -> n!*LaguerreL(n, n+1, -1): # _Peter Luschny_, May 02 2021
%t a[n_] := Sum[k! * Binomial[n, k] * Binomial[2*n+1, k], {k, 0, n}]; Array[a, 20, 0] (* _Amiram Eldar_, May 01 2021 *)
%t Table[(-1)^n * HypergeometricU[-n, 2 + n, -1], {n, 0, 20}] (* _Vaclav Kotesovec_, May 02 2021 *)
%o (PARI) a(n) = sum(k=0, n, k!*binomial(n, k)*binomial(2*n+1, k));
%o (PARI) a(n) = (2*n+1)!*sum(k=0, n, binomial(n, k)/(k+n+1)!);
%o (PARI) a(n) = n!*sum(k=0, n, binomial(2*n+1, k)/(n-k)!);
%o (PARI) a(n) = n!*pollaguerre(n, n+1, -1);
%o (Magma) [Factorial(n)*Evaluate(LaguerrePolynomial(n, n+1), -1): n in [0..40]]; // _G. C. Greubel_, Aug 11 2022
%o (SageMath) [factorial(n)*gen_laguerre(n, n+1, -1) for n in (0..40)] # _G. C. Greubel_, Aug 11 2022
%Y Cf. A000522, A045721, A082545, A152059, A248668, A343830, A343831, A000108.
%K nonn
%O 0,2
%A _Seiichi Manyama_, May 01 2021
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