A270707 - OEIS

%I #11 Sep 07 2018 18:26:38

%S 0,2,3,14,60,349,2310,17772,154224,1494168,15973980,186815386,

%T 2372249880,32503673760,477955820160,7507517217600,125452772867520,

%U 2222130456911520,41587962405967872,820019478835203840,16990772582549183040

%N a(n) = (n+1)!*Sum_{k=0..(n-1)/2}(k!*stirling1(n-k,k+1)*(-1)^(n+1)/(n-k)!/(k+1)!).

%H G. C. Greubel, <a href="/A270707/b270707.txt">Table of n, a(n) for n = 0..250</a>

%F E.g.f.: (log(1/(1-x))+x/(1-x))*(1/(1-x)^x-1)/(x*log(1/(1-x))).

%F a(n) ~ n! * n/log(n) * (1 + (1-gamma)/log(n) + (gamma^2 - 2*gamma + 2 - Pi^2/6)/log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Mar 22 2016

%t Table[(n+1)!*Sum[k!*StirlingS1[n-k, k+1]*(-1)^(n+1)/(n-k)!/(k+1)!, {k, 0, (n-1)/2}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 22 2016 *)

%o (Maxima)

%o makelist((n)!*coeff(taylor((log(1/(1-x))+x/(1-x))*(1/(1-x)^x-1)/(x*log(1/(1-x))),x,0,15),x,n),n,0,15);

%o a(n):=(n+1)!*sum((k)!*stirling1(n-k,k+1)*(-1)^(n+1)/(n-k)!/(k+1)!,k,0,(n-1)/2);

%o (PARI) for(n=0,20, print1((n+1)!*sum(k=0, (n-1)/2,  k!*stirling(n-k, k+1,1)*(-1)^(n+1)/(n-k)!/(k+1)!), ", ")) \\ _G. C. Greubel_, Sep 07 2018

%Y Cf. A048994.

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Mar 22 2016