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A256325
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a(n) = Sum_{k=0..n-1} (n-k)!*exp(-k/2)*M_{k-n,1/2}(k), where M is the Whittaker function.
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2
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0, 0, 1, 5, 24, 136, 933, 7589, 71376, 760796, 9051353, 118784325, 1703388648, 26486926720, 443732646029, 7965563713781, 152504645563072, 3101366761047860, 66753627906345057, 1515914174890163541, 36218232449903567992, 908098606824551207384, 23839591584412453131765
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OFFSET
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0,4
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LINKS
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FORMULA
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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
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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