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 A383678 #44 May 23 2025 03:02:51
%S A383678 1,5,74,1650,48504,1763100,76223664,3817038960,217177416576,
%T A383678 13834411290720,975244141065600,75366122480858880,6335159176892851200,
%U A383678 575442172080117538560,56165570794932257433600,5862137958472255891200000,651508569509254106827161600,76814449419352043102473728000
%N A383678 a(n) = [x^n] Product_{k=0..n} (1 + (2*n+k)*x).
%F A383678 a(n) = A165675(3*n,2*n).
%F A383678 a(n) = Sum_{k=0..n} (k+1) * (2*n)^k * |Stirling1(n+1,k+1)|.
%F A383678 a(n) = (n+1)! * Sum_{k=0..n} (-1)^k * binomial(-2*n,k)/(n+1-k).
%F A383678 a(n) = (3*n)!/(2*n)! * (1 + 2*n * Sum_{k=1..n} 1/(2*n+k)).
%F A383678 a(n) ~ log(3/2) * 3^(3*n + 1/2) * n^(n+1) / (exp(n) * 2^(2*n - 1/2)). - _Vaclav Kotesovec_, May 23 2025
%t A383678 Table[SeriesCoefficient[Product[(1 + (2*n+k)*x), {k, 0, n}], {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, May 23 2025 *)
%o A383678 (PARI) a(n) = sum(k=0, n, (k+1)*(2*n)^k*abs(stirling(n+1, k+1, 1)));
%Y A383678 Column k=2 of A382347.
%Y A383678 Cf. A000142, A165675, A384024.
%K A383678 nonn
%O A383678 0,2
%A A383678 _Seiichi Manyama_, May 18 2025