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A348063 Coefficient of x^2 in expansion of n!* Sum_{k=0..n} binomial(x,k).

%I A348063 #25 Sep 27 2021 18:30:27
%S A348063 1,0,11,5,304,364,15980,34236,1368936,4429656,173699712,771653376,
%T A348063 30605906304,175622947200,7149130156800,50800930272000,
%U A348063 2137822335475200,18241636315507200,796397873127782400,7971407298921830400,361615771356450508800,4168685961862906982400,196587429737202833817600
%N A348063 Coefficient of x^2 in expansion of n!* Sum_{k=0..n} binomial(x,k).
%F A348063 a(n) = a(n-1) + (n-1)^2 * a(n-2) + (-1)^n * (n-2)!.
%F A348063 E.g.f.: (log(1 + x))^2/(2 * (1 - x)).
%F A348063 a(n) ~ n! * log(2)^2 / 2 * (1 + (-1)^n*log(n)/(log(2)^2*n)). - _Vaclav Kotesovec_, Sep 27 2021
%o A348063 (PARI) a(n) = n!*polcoef(sum(k=2, n, binomial(x, k)), 2);
%o A348063 (PARI) a(n) = if(n<2, 0, a(n-1)+(n-1)^2*a(n-2)+(-1)^n*(n-2)!);
%o A348063 (PARI) N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^2/(2*(1-x))))
%o A348063 (Python)
%o A348063 from sympy.abc import x
%o A348063 from sympy import ff, expand
%o A348063 def A348063(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**2) for k in range(2,n+1)) # _Chai Wah Wu_, Sep 27 2021
%Y A348063 Column k=2 of A190782.
%Y A348063 Cf. A000399, A028339, A054651, A347987, A348064, A348065, A348068.
%K A348063 nonn
%O A348063 2,3
%A A348063 _Seiichi Manyama_, Sep 26 2021