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

0,4

Links

Robert Israel, Table of n, a(n) for n = 0..364

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

Conjecture verified using the differential equation y + (124*x - 31)*y' + (240*x^2 - 120*x + 15)*y'' + (64*x^3 - 48*x^2 + 12*x - 1)*y''' = 0 satisfied by the e.g.f. - Robert Israel, Mar 12 2026

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

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

Prog

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

Crossrefs

Column k=2 of A290319.

Sequence in context: A218368 A123816 A273921 * A182607 A218284 A218634

Adjacent sequences: A383697 A383698 A383699 * A383701 A383702 A383703

Keyword

nonn

Author

Seiichi Manyama, May 06 2025

Status

approved