A367333 a(n) = 27^n * Sum_{k=0..n} binomial(-1/3, k)^2.

1, 30, 846, 23430, 643635, 17601732, 480016620, 13065872292, 355170348720, 9644965082940, 261716257738980, 7097365769203260, 192376104782028120, 5212313820585819540, 141177183151026767580, 3822747528826291049460, 103486045894075138514445

Offset: 0

Author

Vaclav Kotesovec, Nov 14 2023

Keywords

• nonn

Comments

Compare with A358365: Sum_{k>=0} binomial(-1/3, k)^2 converges, but Sum_{k>=0} binomial(-1/2, k)^2 diverges.

In general, for m>2, Sum_{k>=0} binomial(-1/m,k)^2 = Gamma(1 - 2/m) / Gamma(1 - 1/m)^2.

Crossrefs

Cf. A358365, A367330, A367331, A367332.

Programs

Mathematica

Table[27^n*Sum[Binomial[-1/3, k]^2, {k, 0, n}], {n, 0, 16}]

Formula

a(n) ~ Gamma(1/3)^3 * 3^(3*n+1) / (4*Pi^2).