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 A384060 #18 May 19 2025 04:50:49
%S A384060 1,4,82,3024,162154,11438280,1001454024,104777127616,12755141675754,
%T A384060 1771354690734420,276386332002204450,47870892086756660064,
%U A384060 9113932961179205496744,1891845220489637114281216,425240943851497448491619600,102899751348092720847554016000
%N A384060 a(n) = [x^n] Product_{k=0..n} 1/(1 - k*x)^4.
%C A384060 In general, for m>=1, [x^n] Product_{k=0..n} 1/(1 - k*x)^m ~ (m+1)^((m+1)*n + (m-1)/2) * n^(n - 1/2) / (sqrt(2*Pi*(1-w)) * exp(n) * (m+1-m*w)^n * m^(m*(n + 1/2)) * w^(m*n + (m-1)/2)), where w = -LambertW(-(m+1)*exp(-(m+1)/m)/m).
%C A384060 The general formula is valid even for m=n, where after modifications we get the formula for A351508.
%F A384060 a(n) = Sum_{i, j, k, l>=0 and i+j+k+l=n} Stirling2(i+n,n) * Stirling2(j+n,n) * Stirling2(k+n,n) * Stirling2(l+n,n). - _Seiichi Manyama_, May 18 2025
%F A384060 a(n) ~ 5^(5*n + 3/2) * n^(n - 1/2) / (sqrt(Pi*(1-w)) * 2^(8*n + 9/2) * exp(n) * (5 - 4*w)^n * w^(4*n + 3/2)), where w = -LambertW(-5*exp(-5/4)/4) = 0.7857872456211833502961937693700363613539172187... - _Vaclav Kotesovec_, May 18 2025
%t A384060 Table[SeriesCoefficient[Product[1/(1-k*x)^4, {k, 1, n}], {x, 0, n}], {n, 0, 15}]
%Y A384060 Cf. A007820 (m=1), A350376 (m=2), A383862 (m=3), A351508 (m=n).
%Y A384060 Cf. A384031.
%K A384060 nonn
%O A384060 0,2
%A A384060 _Vaclav Kotesovec_, May 18 2025