A383700 Coefficient of x^2 in expansion of (x+1) * (x+5) * ... * (x+4*n-3).

0, 0, 1, 15, 254, 5130, 122119, 3365089, 105599276, 3722336388, 145717348221, 6275071262691, 294890141047050, 15020233818893550, 824373714907080675, 48505985450168267925, 3046201904592803410200, 203381159927362120499400, 14385952383695375700375225

Offset: 0

Author

Seiichi Manyama, May 06 2025

Keywords

• nonn

Crossrefs

Column k=2 of A290319.

Programs

Maple

A383700 := proc(n)
    add((4*n-3)^(k-2) * 4^(n-k) * binomial(k,2) * stirling1(n,k),k=2..n) ;
end proc:
seq(A383700(n),n=0..60) ; # R. J. Mathar, May 07 2025

PARI

a(n) = polcoef(prod(k=0, n-1, x+4*k+1), 2);

Formula

a(n) = Sum_{k=2..n} 4^(n-k) * binomial(k,2) * |Stirling1(n,k)|.

a(n) = Sum_{k=2..n} (4*n-3)^(k-2) * 4^(n-k) * binomial(k,2) * Stirling1(n,k).

E.g.f.: f(x) * log(f(x))^2 / 2, where f(x) = 1/(1 - 4*x)^(1/4).

Conjecture D-finite with recurrence a(n) +3*(-4*n+7)*a(n-1) +(48*n^2-216*n+247)*a(n-2) -(4*n-11)^3*a(n-3)=0. - R. J. Mathar, May 07 2025

a(n) ~ sqrt(Pi) * 2^(2*n - 5/2) * n^(n - 1/4) * log(n)^2 / (Gamma(1/4) * exp(n)) * (1 + (2*gamma + Pi + 6*log(2))/log(n)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function. - Vaclav Kotesovec, May 12 2025