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 A384017 #42 May 19 2025 04:35:45
%S A384017 1,5,100,3510,177370,11732175,960453825,93791830160,10644367637490,
%T A384017 1376936603007075,200002385378370350,32233130183113838550,
%U A384017 5708169533474858008905,1101836121788665346133960,230256048227047074266497400,51791322674249971562728368000
%N A384017 a(n) = [x^n] Product_{k=0..n} (1 + k*x)^5.
%C A384017 From _Vaclav Kotesovec_, May 19 2025: (Start)
%C A384017 In general, for m > 1, [x^n] Product_{k=0..n} (1 + k*x)^m ~ m^(m*(n + 1/2)) * w^(m*n + (m+1)/2) * n^(n - 1/2) / (sqrt(2*Pi*(w-1)) * exp(n) * (m-1)^((m-1)*n + (m+1)/2) * (m*w-m+1)^n), where w = -LambertW(-1,-(m-1)*exp(-(m-1)/m)/m).
%C A384017 The general formula is valid even for m=n, where after modifications we get the formula for A351507. (End)
%F A384017 a(n) = Sum_{0<=i, j, k, l, m<=n and i+j+k+l+m=4*n} |Stirling1(n+1,i+1) * Stirling1(n+1,j+1) * Stirling1(n+1,k+1) * Stirling1(n+1,l+1) * Stirling1(n+1,m+1)|.
%F A384017 a(n) ~ 5^(5*n + 5/2) * w^(5*n+3) * n^(n - 1/2) / (2^(8*n + 13/2) * sqrt(Pi*(w-1)) * exp(n) * (5*w-4)^n), where w = -LambertW(-1,-4*exp(-4/5)/5) = 1.2308422097842590367678406745433500325966... - _Vaclav Kotesovec_, May 19 2025
%t A384017 Table[SeriesCoefficient[Product[(1+k*x)^5, {k, 1, n}], {x, 0, n}], {n, 0, 15}] (* _Vaclav Kotesovec_, May 19 2025 *)
%o A384017 (PARI) a(n) = polcoef(prod(k=1, n, 1+k*x)^5, n);
%Y A384017 Cf. A000142 (m=1), A129256 (m=2), A384012 (m=3), A384031 (m=4), A351507 (m=n).
%K A384017 nonn
%O A384017 0,2
%A A384017 _Seiichi Manyama_, May 18 2025