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

0,3

Links

Table of n, a(n) for 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

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));

Mathematica

a[n_] := n n! HypergeometricPFQ[{1 - n, 1 - n}, {2, 2}, 1]; a[0] := 1;
Table[a[n], {n, 0, 20}]

Crossrefs

Cf. A271703, A111596, A344050.

Sequence in context: A208813 A112698 A234953 * A025168 A084358 A050351

Adjacent sequences: A344048 A344049 A344050 * A344052 A344053 A344054

Keyword

nonn

Author

Peter Luschny, May 10 2021

Status

approved