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Coefficient of x^2 in expansion of n!* Sum_{k=0..n} binomial(x,k).
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%I #25 Sep 27 2021 18:30:27

%S 1,0,11,5,304,364,15980,34236,1368936,4429656,173699712,771653376,

%T 30605906304,175622947200,7149130156800,50800930272000,

%U 2137822335475200,18241636315507200,796397873127782400,7971407298921830400,361615771356450508800,4168685961862906982400,196587429737202833817600

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

%F a(n) = a(n-1) + (n-1)^2 * a(n-2) + (-1)^n * (n-2)!.

%F E.g.f.: (log(1 + x))^2/(2 * (1 - x)).

%F a(n) ~ n! * log(2)^2 / 2 * (1 + (-1)^n*log(n)/(log(2)^2*n)). - _Vaclav Kotesovec_, Sep 27 2021

%o (PARI) a(n) = n!*polcoef(sum(k=2, n, binomial(x, k)), 2);

%o (PARI) a(n) = if(n<2, 0, a(n-1)+(n-1)^2*a(n-2)+(-1)^n*(n-2)!);

%o (PARI) N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^2/(2*(1-x))))

%o (Python)

%o from sympy.abc import x

%o from sympy import ff, expand

%o 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 Column k=2 of A190782.

%Y Cf. A000399, A028339, A054651, A347987, A348064, A348065, A348068.

%K nonn

%O 2,3

%A _Seiichi Manyama_, Sep 26 2021