A344051 a(n) = Sum_{k=0..n} binomial(n, k)*|Lah(n, k)|. Binomial convolution of the unsigned Lah numbers A271703.
1, 1, 5, 37, 361, 4301, 60001, 954325, 16984577, 333572041, 7151967181, 165971975621, 4139744524345, 110333560295557, 3126749660583641, 93819198847833061, 2969676820062708481, 98843743790129998865, 3449675368718647501717, 125921086600579132143781, 4796519722094585691925961
Offset: 0
Keywords
Programs
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Maple
aList := proc(len) local lah; lah := (n, k) -> `if`(n = k, 1, binomial(n-1, k-1)*n!/k!): seq(add(binomial(n, k)*lah(n, k), k = 0..n), n = 0..len-1) end: lprint(aList(22));
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Mathematica
a[n_] := n n! HypergeometricPFQ[{1 - n, 1 - n}, {2, 2}, 1]; a[0] := 1; Table[a[n], {n, 0, 20}]
Formula
a(n) = n * n! * hypergeom([1 - n, 1 - n], [2, 2], 1) for n >= 1.
D-finite with recurrence +16*n*a(n) +6*(-8*n^2+5*n-1)*a(n-1) +(48*n^3-266*n^2+407*n-167)*a(n-2) +(-16*n^4+106*n^3-219*n^2+108*n+93)*a(n-3) +(n-4)*(2*n^3-13*n^2+16*n+25)*a(n-4) -(n-5)*(n-4)^3*a(n-5)=0. - R. J. Mathar, Jul 27 2022
a(n) ~ n^(n - 1/2) / (sqrt(6*Pi) * exp(n - 3*n^(2/3) + n^(1/3) - 1/3)) * (1 + 31/(54*n^(1/3))). - Vaclav Kotesovec, Apr 27 2024