%I #10 Feb 24 2021 08:20:00
%S 0,0,1,5,24,136,933,7589,71376,760796,9051353,118784325,1703388648,
%T 26486926720,443732646029,7965563713781,152504645563072,
%U 3101366761047860,66753627906345057,1515914174890163541,36218232449903567992,908098606824551207384,23839591584412453131765
%N a(n) = Sum_{k=0..n-1} (n-k)!*exp(-k/2)*M_{k-n,1/2}(k), where M is the Whittaker function.
%H G. C. Greubel, <a href="/A256325/b256325.txt">Table of n, a(n) for n = 0..250</a>
%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F a(n) = Sum_{k=0..n-1} k*(n-k)!*hypergeom([k-n+1],[2],-k).
%F a(n) = Sum_{k=0..n-1}(Sum_{j=0.. n-k}((n-k-j)!*C(n-k,j)*C(n-k-1,j-1)*k^j)).
%F a(n) = Sum_{k=0..n-1} (n-k-1)* k! * LaguerreL(k, 1, k-n+1). - _G. C. Greubel_, Feb 23 2021
%p a := n -> add(exp(-k/2)*WhittakerM(-(n-k),1/2,k)*(n-k)!,k=0..n-1):
%p seq(round(evalf(a(n),64)), n=0..22);
%p # Alternatively:
%p a := n -> add(k*(n-k)!*hypergeom([k-n+1],[2],-k),k=0..n-1):
%p seq(simplify(a(n)), n=0..22);
%t Table[Sum[(n-k-1)*k!*LaguerreL[k, 1, k-n+1], {k,0,n-1}], {n,0,30}] (* _G. C. Greubel_, Feb 23 2021 *)
%o (Sage) [sum( (n-k-1)*factorial(k)*gen_laguerre(k, 1, k-n+1) for k in (0..n-1) ) for n in (0..30)] # _G. C. Greubel_, Feb 23 2021
%o (Magma) [n eq 0 select 0 else (&+[(n-k-1)*Factorial(k)*Evaluate( LaguerrePolynomial(k, 1), k-n+1): k in [0..n-1]]): n in [0..30]]; // _G. C. Greubel_, Feb 23 2021
%Y Cf. A253286.
%K nonn,easy
%O 0,4
%A _Peter Luschny_, Mar 24 2015