A364658 Numerators of coefficients in expansion of (1 + x)^(2/3).

1, 2, -1, 4, -7, 14, -91, 208, -494, 10868, -27170, 69160, -535990, 1401820, -3704810, 29638480, -79653415, 215532770, -5280552865, 14452039420, -39743108405, 329300041070, -913059204785, 2540686482880, -21278249294120, 59579098023536, -167279775219928, 12713262916714528

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1401

Formula

a(n) / A067623(n) ~ 2*(-1)^(n+1) / (3*Gamma(1/3)*n^(5/3)). - Vaclav Kotesovec, Jan 06 2026

Example

(1 + x)^(2/3) = 1 + 2*x/3 - x^2/9 + 4*x^3/81 - 7*x^4/243 + 14*x^5/729 - 91*x^6/6561 + ...
Coefficients are 1, 2/3, -1/9, 4/81, -7/243, 14/729, -91/6561, ...

Maple

f:= n -> numer(2*(-1)^(n+1)*pochhammer(1/3, n-1)/(3*n!)):
map(f, [$0..30]); # Robert Israel, Jan 05 2026

Mathematica

nmax = 27; CoefficientList[Series[(1 + x)^(2/3), {x, 0, nmax}], x] // Numerator
Table[Binomial[2/3, n], {n, 0, 27}] // Numerator

Prog

(PARI) my(x='x+O('x^30)); apply(numerator, Vec((1 + x)^(2/3))) \\ Michel Marcus, Aug 02 2023

Crossrefs

Denominators are A067623.

Cf. A002596, A067622, A127974, A161200, A364661.

Sequence in context: A378343 A275778 A007839 * A184345 A045625 A294501

Adjacent sequences: A364655 A364656 A364657 * A364659 A364660 A364661

Keyword

sign,frac

Author

Ilya Gutkovskiy, Aug 01 2023

Status

approved