A178185 Numerator of Sum_{k=1..n} 1/2^((k^2 + 3*k)/2).

1, 9, 145, 4641, 297025, 38019201, 9732915457, 4983252713985, 5102850779120641, 10450638395639072769, 42805814868537642061825, 350665235403060363770470401, 5745299216843741000015387049985

Offset: 1

Author

Artur Jasinski, May 21 2010

Keywords

• nonn
• frac

Comments

Series of the kind m^((k^2 + 3*k)/2) from k=1 to n were studied by Bernoulli and Euler.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..80

Crossrefs

Cf. A178184-A178193.

Cf. A036442 (denominators).

Programs

Mathematica

aa = {}; m = 1/2; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, Numerator[sum]], {n, 1, 20}]; aa
Numerator[Table[Sum[1/2^((k^2 + 3*k)/2), {k, 1, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Apr 10 2024 *)

PARI

a(n) = numerator(sum(k=1, n, (1/2)^((k^2+3*k)/2))); \\ Michel Marcus, Sep 09 2013

Formula

a(n) = 2^(n+1)*a(n-1) + 1, a(1) = 1. - Alexandre Herrera, Mar 23 2024

a(n) ~ c * A036442(n+1) = c * 2^(n*(n+3)/2), where c = 2^(1/8) * EllipticTheta[2, 0, 1/Sqrt[2]] - 3 [in Mathematica] = 2^(1/8) * JacobiTheta2(0, 1/sqrt(2)) - 3 [in Maple] = 0.2832651213103077325876855404508588684521230759134794956... - Vaclav Kotesovec, Apr 10 2024