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 A384024 #19 May 18 2025 09:58:09
%S A384024 1,3,26,342,5944,127860,3272688,97053936,3270729600,123418922400,
%T A384024 5154170774400,235977273544320,11752173128586240,632474276804697600,
%U A384024 36576553723886131200,2261980049125982976000,148956705206745595084800,10406288081667512679321600,768701832940487804295168000
%N A384024 a(n) = [x^n] Product_{k=0..n} (1 + (n+k)*x).
%F A384024 a(n) ~ n! * log(2) * 4^n * sqrt(n/Pi).
%F A384024 a(n) ~ log(2) * 2^(2*n + 1/2) * n^(n+1) / exp(n).
%F A384024 From _Seiichi Manyama_, May 18 2025: (Start)
%F A384024 a(n) = A165675(2*n,n).
%F A384024 a(n) = Sum_{k=0..n} (k+1) * n^k * |Stirling1(n+1,k+1)|.
%F A384024 a(n) = (n+1)! * Sum_{k=0..n} (-1)^k * binomial(-n,k)/(n+1-k).
%F A384024 a(n) = (2*n)!/n! * (1 + n * Sum_{k=1..n} 1/(n+k)). (End)
%t A384024 Table[SeriesCoefficient[Product[1 + (n+k)*x, {k, 0, n}], {x, 0, n}], {n, 0, 20}]
%o A384024 (PARI) a(n) = sum(k=0, n, (k+1)*n^k*abs(stirling(n+1, k+1, 1))); \\ _Seiichi Manyama_, May 18 2025
%Y A384024 Central terms of triangle A165675.
%Y A384024 Cf. A000407, A201546, A383869.
%K A384024 nonn
%O A384024 0,2
%A A384024 _Vaclav Kotesovec_, May 17 2025