A344051 - OEIS

%I #11 Apr 27 2024 04:50:16

%S 1,1,5,37,361,4301,60001,954325,16984577,333572041,7151967181,

%T 165971975621,4139744524345,110333560295557,3126749660583641,

%U 93819198847833061,2969676820062708481,98843743790129998865,3449675368718647501717,125921086600579132143781,4796519722094585691925961

%N a(n) = Sum_{k=0..n} binomial(n, k)*|Lah(n, k)|. Binomial convolution of the unsigned Lah numbers A271703.

%F a(n) = n * n! * hypergeom([1 - n, 1 - n], [2, 2], 1] for n >= 1.

%F 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

%F 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

%p aList := proc(len) local lah;

%p lah := (n, k) -> `if`(n = k, 1, binomial(n-1, k-1)*n!/k!):

%p seq(add(binomial(n, k)*lah(n, k), k = 0..n), n = 0..len-1) end:

%p lprint(aList(22));

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

%t Table[a[n], {n, 0, 20}]

%Y Cf. A271703, A111596, A344050.

%K nonn

%O 0,3

%A _Peter Luschny_, May 10 2021