This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383700 #16 May 12 2025 03:51:17
%S A383700 0,0,1,15,254,5130,122119,3365089,105599276,3722336388,145717348221,
%T A383700 6275071262691,294890141047050,15020233818893550,824373714907080675,
%U A383700 48505985450168267925,3046201904592803410200,203381159927362120499400,14385952383695375700375225
%N A383700 Coefficient of x^2 in expansion of (x+1) * (x+5) * ... * (x+4*n-3).
%F A383700 a(n) = Sum_{k=2..n} 4^(n-k) * binomial(k,2) * |Stirling1(n,k)|.
%F A383700 a(n) = Sum_{k=2..n} (4*n-3)^(k-2) * 4^(n-k) * binomial(k,2) * Stirling1(n,k).
%F A383700 E.g.f.: f(x) * log(f(x))^2 / 2, where f(x) = 1/(1 - 4*x)^(1/4).
%F A383700 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
%F A383700 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
%p A383700 A383700 := proc(n)
%p A383700 add((4*n-3)^(k-2) * 4^(n-k) * binomial(k,2) * stirling1(n,k),k=2..n) ;
%p A383700 end proc:
%p A383700 seq(A383700(n),n=0..60) ; # _R. J. Mathar_, May 07 2025
%o A383700 (PARI) a(n) = polcoef(prod(k=0, n-1, x+4*k+1), 2);
%Y A383700 Column k=2 of A290319.
%K A383700 nonn
%O A383700 0,4
%A A383700 _Seiichi Manyama_, May 06 2025