A256325 a(n) = Sum_{k=0..n-1} (n-k)!*exp(-k/2)*M_{k-n,1/2}(k), where M is the Whittaker function.

0, 0, 1, 5, 24, 136, 933, 7589, 71376, 760796, 9051353, 118784325, 1703388648, 26486926720, 443732646029, 7965563713781, 152504645563072, 3101366761047860, 66753627906345057, 1515914174890163541, 36218232449903567992, 908098606824551207384, 23839591584412453131765

Offset: 0

Author

Peter Luschny, Mar 24 2015

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A253286.

Programs

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

Maple

a := n -> add(exp(-k/2)*WhittakerM(-(n-k),1/2,k)*(n-k)!,k=0..n-1):
seq(round(evalf(a(n),64)), n=0..22);
# Alternatively:
a := n -> add(k*(n-k)!*hypergeom([k-n+1],[2],-k),k=0..n-1):
seq(simplify(a(n)), n=0..22);

Mathematica

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 *)

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

Formula

a(n) = Sum_{k=0..n-1} k*(n-k)!*hypergeom([k-n+1],[2],-k).

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)).

a(n) = Sum_{k=0..n-1} (n-k-1)* k! * LaguerreL(k, 1, k-n+1). - G. C. Greubel, Feb 23 2021