cp's OEIS Frontend

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

%I A387205 #18 Aug 29 2025 11:08:57
%S A387205 1,2,3,7,23,98,514,3204,23148,190224,1752336,17886240,200377440,
%T A387205 2444446080,32256800640,457822229760,6954511737600,112579862169600,
%U A387205 1934780446771200,35181735469977600,674855347635302400,13618752053114880000,288426695123589120000,6396478234890670080000
%N A387205 a(n) = (n - 1)!*(2 + Harmonic(n - 1)) if n >= 1, and a(0) = 1.
%H A387205 Paolo Xausa, <a href="/A387205/b387205.txt">Table of n, a(n) for n = 0..400</a>
%F A387205 a(n) = 2*|Stirling1(n, 1)| + |Stirling1(n, 2)| for n >= 1.
%F A387205 a(n) = n! * [x^n] Laguerre(2, log(1 - x)).
%F A387205 a(n) = Gamma(n)*(PolyGamma(n) + EulerGamma + 2) for n >= 1.
%F A387205 Conjecture: Maple returns the exponential series expansion at x = 0:
%F A387205 a(n) = n! * [x^n] (1 + tau + (log(x - 1)^2 - (tau + 4)*log(x - 1) - Pi^2)/2) where tau = 2*Pi*I.
%p A387205 a := n -> if n = 0 then 1 else (n-1)!*(2 + harmonic(n-1)) fi:
%p A387205 ser := series(LaguerreL(2, log(1 - x)), x, 24): a := n -> n! * coeff(ser, x, n):
%p A387205 seq(a(n), n = 0..23);
%t A387205 A387205[n_] := If[n == 0, 1, (n - 1)!*(2 + HarmonicNumber[n - 1])];
%t A387205 Array[A387205, 25, 0] (* _Paolo Xausa_, Aug 29 2025 *)
%o A387205 (PARI) a(n) = if (n>0, (n-1)!*(2 + sum(i=1, n-1, 1/i)), 1); \\ _Michel Marcus_, Aug 27 2025
%Y A387205 Cf. A387152 (column 2), A001008, A130534.
%K A387205 nonn,new
%O A387205 0,2
%A A387205 _Peter Luschny_, Aug 27 2025