A343849 a(n) = Sum_{k = 0..n} n! * LaguerreL(n, -k).
1, 3, 23, 294, 5194, 116620, 3175717, 101696700, 3745365444, 155975005998, 7247927859875, 371803988506742, 20870023274690966, 1272424816703533792, 83736949788656865729, 5916106654032037435800, 446636583718649775483144, 35882981062654529341219962, 3056767877633271802374850239
Offset: 0
Keywords
Programs
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Mathematica
a[n_] := Sum[n! LaguerreL[n, -k], {k, 0, n}]; Table[a[n], {n, 0, 18}] -
PARI
a(n) = n!*sum(m=0, n, sum(j=0, n, binomial(n, j) * m^j / j!)) for(n=0, 18, print(a(n)))
Formula
a(n) = (-1)^n * Sum_{k=0..n} KummerU(-n, 1, -k).
a(n) = n! * Sum_{m=0..n} Sum_{j=0..n} binomial(n, j) * m^j / j!.
a(n) ~ exp(n/phi - n) * phi^(2*n+1) * n^n / (5^(1/4) * (1 - exp(-1/phi))), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 09 2021