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 A361291 #28 Mar 14 2023 12:55:56
%S A361291 1,6,57,820,16105,402234,12204241,435984840,17927094321,833994048910,
%T A361291 43309534450633,2483526865641276,155867505885345241,
%U A361291 10627079738421409410,782175399728156197665,61812037545704964935440,5220088150634922700769761,469168161404536131943150998
%N A361291 a(n) = ((2*n + 1)^n - 1)/(2*n).
%C A361291 This sequence is of the form (k^n - 1)/(k - 1) with k = 2*n + 1. See crossrefs in A218722 for other sequences of the same form.
%F A361291 a(n) = Sum_{i=0..n-1} A005408(n)^i.
%F A361291 a(n) = n! * [x^n] exp(x)*(exp(2*n*x) - 1)/(2*n).
%F A361291 a(n) = n! * [x^n] exp((n+1)*x)*sinh(n*x)/n.
%F A361291 Limit_{n->oo} a(n+1)/(n*a(n)) = 2*e.
%F A361291 Limit_{n->oo} (a(n+1)/a(n) - a(n)/a(n-1)) = 2*e.
%t A361291 Table[((2n+1)^n-1)/(2n),{n,20}]
%o A361291 (Python)
%o A361291 def A361291(n): return (((n<<1)+1)**n-1)//(n<<1) # _Chai Wah Wu_, Mar 14 2023
%Y A361291 Cf. A005408, A019762 (2*e), A051129, A218722.
%Y A361291 Cf. A000169, A000312, A038057, A052746, A062971, A213236.
%K A361291 nonn,easy
%O A361291 1,2
%A A361291 _Stefano Spezia_, Mar 12 2023