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 A293574 #9 Oct 16 2017 04:27:29
%S A293574 1,2,11,82,787,9476,139134,2422218,48824675,1118286172,28679699578,
%T A293574 814027423892,25330145185646,857375286365768,31360145331198428,
%U A293574 1232586016712594010,51805909208539809315,2318588202311267591852,110085368092924083334626,5526615354023679440754396,292501304641192746350100410
%N A293574 a(n) = Sum_{k=0..n} n^(n-k)*binomial(n+k-1,k).
%C A293574 a(n) is the n-th term of the main diagonal of iterated partial sums array of powers of n (see example).
%F A293574 a(n) = [x^n] 1/((1 - x)^n*(1 - n*x)).
%F A293574 a(n) ~ exp(1) * n^n. - _Vaclav Kotesovec_, Oct 16 2017
%e A293574 For n = 2 we have:
%e A293574 ----------------------------
%e A293574 0 1 [2] 3 4 5
%e A293574 ----------------------------
%e A293574 1, 2, 4, 8, 16, 32, ... A000079 (powers of 2)
%e A293574 1, 3, 7, 15, 31, 63, ... A126646 (partial sums of A000079)
%e A293574 1, 4, [11], 26, 57, 120, ... A000295 (partial sums of A126646)
%e A293574 ----------------------------
%e A293574 therefore a(2) = 11.
%t A293574 Join[{1}, Table[Sum[n^(n - k) Binomial[n + k - 1, k], {k, 0, n}], {n, 1, 20}]]
%t A293574 Table[SeriesCoefficient[1/((1 - x)^n (1 - n x)), {x, 0, n}], {n, 0, 20}]
%t A293574 Join[{1, 2}, Table[n^(2 n)/(n - 1)^n - Binomial[2 n, n + 1] Hypergeometric2F1[1, 2 n + 1, n + 2, 1/n]/n, {n, 2, 20}]]
%o A293574 (PARI) a(n) = sum(k=0, n, n^(n-k)*binomial(n+k-1,k)); \\ _Michel Marcus_, Oct 12 2017
%Y A293574 Cf. A000312, A001700, A031973, A032443, A100192, A100193.
%K A293574 nonn
%O A293574 0,2
%A A293574 _Ilya Gutkovskiy_, Oct 12 2017