A361291 a(n) = ((2*n + 1)^n - 1)/(2*n).

1, 6, 57, 820, 16105, 402234, 12204241, 435984840, 17927094321, 833994048910, 43309534450633, 2483526865641276, 155867505885345241, 10627079738421409410, 782175399728156197665, 61812037545704964935440, 5220088150634922700769761, 469168161404536131943150998

Offset: 1

Author

Stefano Spezia, Mar 12 2023

Keywords

• nonn
• easy

Comments

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.

Crossrefs

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

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

Programs

Mathematica

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

Python

def A361291(n): return (((n<<1)+1)**n-1)//(n<<1) # Chai Wah Wu, Mar 14 2023

Formula

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

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

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

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

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