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a(n) = ((2*n + 1)^n - 1)/(2*n).
2

%I #28 Mar 14 2023 12:55:56

%S 1,6,57,820,16105,402234,12204241,435984840,17927094321,833994048910,

%T 43309534450633,2483526865641276,155867505885345241,

%U 10627079738421409410,782175399728156197665,61812037545704964935440,5220088150634922700769761,469168161404536131943150998

%N a(n) = ((2*n + 1)^n - 1)/(2*n).

%C 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 a(n) = Sum_{i=0..n-1} A005408(n)^i.

%F a(n) = n! * [x^n] exp(x)*(exp(2*n*x) - 1)/(2*n).

%F a(n) = n! * [x^n] exp((n+1)*x)*sinh(n*x)/n.

%F Limit_{n->oo} a(n+1)/(n*a(n)) = 2*e.

%F Limit_{n->oo} (a(n+1)/a(n) - a(n)/a(n-1)) = 2*e.

%t Table[((2n+1)^n-1)/(2n),{n,20}]

%o (Python)

%o def A361291(n): return (((n<<1)+1)**n-1)//(n<<1) # _Chai Wah Wu_, Mar 14 2023

%Y Cf. A005408, A019762 (2*e), A051129, A218722.

%Y Cf. A000169, A000312, A038057, A052746, A062971, A213236.

%K nonn,easy

%O 1,2

%A _Stefano Spezia_, Mar 12 2023