A387205 a(n) = (n - 1)!*(2 + Harmonic(n - 1)) if n >= 1, and a(0) = 1.

1, 2, 3, 7, 23, 98, 514, 3204, 23148, 190224, 1752336, 17886240, 200377440, 2444446080, 32256800640, 457822229760, 6954511737600, 112579862169600, 1934780446771200, 35181735469977600, 674855347635302400, 13618752053114880000, 288426695123589120000, 6396478234890670080000

Offset: 0

Author

Peter Luschny, Aug 27 2025

Keywords

• nonn
• new

Links

• Paolo Xausa, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A387152 (column 2), A001008, A130534.

Programs

Maple

a := n -> if n = 0 then 1 else (n-1)!*(2 + harmonic(n-1)) fi:
ser := series(LaguerreL(2, log(1 - x)), x, 24): a := n -> n! * coeff(ser, x, n):
seq(a(n), n = 0..23);

Mathematica

A387205[n_] := If[n == 0, 1, (n - 1)!*(2 + HarmonicNumber[n - 1])];
Array[A387205, 25, 0] (* Paolo Xausa, Aug 29 2025 *)

PARI

a(n) = if (n>0, (n-1)!*(2 + sum(i=1, n-1, 1/i)), 1); \\ Michel Marcus, Aug 27 2025

Formula

a(n) = 2*|Stirling1(n, 1)| + |Stirling1(n, 2)| for n >= 1.

a(n) = n! * [x^n] Laguerre(2, log(1 - x)).

a(n) = Gamma(n)*(PolyGamma(n) + EulerGamma + 2) for n >= 1.

Conjecture: Maple returns the exponential series expansion at x = 0:

a(n) = n! * [x^n] (1 + tau + (log(x - 1)^2 - (tau + 4)*log(x - 1) - Pi^2)/2) where tau = 2*Pi*I.