A234144 a(n) = numerator of Sum_(k=1..n) 1/(2*k - 1)^n.

0, 1, 10, 3527, 123296356, 3115356499043, 1733194364791766081374, 376470435881775086250915790503469, 16952748458548438370767527584555153032, 90548635884513844033505877600764150558334149264809109

Offset: 0

Author

Jean-François Alcover, Dec 20 2013

Keywords

• nonn
• frac
• easy

Comments

The sequence A234144(n)/A234145(n) is Theta(n, n), as defined by Wolfdieter Lang.

Links

• Robert Israel, Table of n, a(n) for n = 0..35
• Wolfdieter Lang, Theta(k, n), k-family of rational sequences and limits.

Crossrefs

Cf. A164655, A164656, A234145 (denominators).

Programs

Maple

f:= proc(n) local k; numer(add(1/(2*k-1)^n,k=1..n)); end proc:
map(f, [$0..10]); # Robert Israel, Nov 05 2023

Mathematica

a[n_] := Sum[1/(2*k-1)^n, {k, 1, n}] // Numerator; Table[a[n], {n, 0, 10}]

Formula

a(n) = numerator of (2^n*Zeta(n) - Zeta(n) - Zeta(n, n+1/2))/2^n.

a(n) = numerator of ((-1/2)^n*(PolyGamma(n-1, 1/2) - PolyGamma(n-1, n+1/2)))/(n-1)!.

A234144(n) / A234145(n) ~ 1.