A377238 Expansion of 1/(1 - 9*x/(1 - 9*x)^(1/3))^(1/3).

1, 3, 27, 288, 3267, 38232, 456030, 5511726, 67275765, 827544276, 10243778238, 127471232682, 1593323199828, 19992464465031, 251700257824749, 3178185350410386, 40235213754593325, 510553935341621550, 6492029680359751572, 82705296277037467728, 1055413999621027732986

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..891

Formula

a(n) = 9^n * Sum_{k=0..n} (-1)^k * binomial(-1/3,k) * binomial(n-2*k/3-1,n-k).

D-finite with recurrence: 26244*(9*n + 2)*(9*n + 8)*(9*n + 14)*a(n) - 4374*(2916*n^3 + 17901*n^2 + 34011*n + 20146)*a(n + 1) + 729*(4941*n^3 + 45171*n^2 + 135576*n + 132236)*a(n + 2) - 243*(2565*n^3 + 28431*n^2 + 108444*n + 139828)*a(n + 3) + 162*(540*n^3 + 6453*n^2 + 26740*n + 38412)*a(n + 4) - 27*(399*n^3 + 5313*n^2 + 23722*n + 35688)*a(n + 5) + 18*(53*n + 248)*(n + 6)*(n + 5)*a(n + 6) - 6*(n + 7)*(n + 6)*(8*n + 43)*a(n + 7) + (n + 8)*(n + 7)*(n + 6)*a(n + 8) = 0. - Robert Israel, Feb 24 2026

Maple

f:= rectoproc({26244*(9*n + 2)*(9*n + 8)*(9*n + 14)*a(n) - 4374*(2916*n^3 + 17901*n^2 + 34011*n + 20146)*a(n + 1) + 729*(4941*n^3 + 45171*n^2 + 135576*n + 132236)*a(n + 2) - 243*(2565*n^3 + 28431*n^2 + 108444*n + 139828)*a(n + 3) + 162*(540*n^3 + 6453*n^2 + 26740*n + 38412)*a(n + 4) - 27*(399*n^3 + 5313*n^2 + 23722*n + 35688)*a(n + 5) + 18*(53*n + 248)*(n + 6)*(n + 5)*a(n + 6) - 6*(n + 7)*(n + 6)*(8*n + 43)*a(n + 7) + (n + 8)*(n + 7)*(n + 6)*a(n + 8), a(0) = 1, a(1) = 3, a(2) = 27, a(3) = 288, a(4) = 3267, a(5) = 38232, a(6) = 456030, a(7) = 5511726} , a(n), remember):
map(f, [$0..40]); # Robert Israel, Feb 24 2026

Prog

(PARI) a(n) = 9^n*sum(k=0, n, (-1)^k*binomial(-1/3, k)*binomial(n-2*k/3-1, n-k));

Crossrefs

Cf. A011782, A377237.

Cf. A377240.

Sequence in context: A011544 A127503 A214363 * A204821 A365156 A372203

Adjacent sequences: A377235 A377236 A377237 * A377239 A377240 A377241

Keyword

nonn

Author

Seiichi Manyama, Oct 21 2024

Status

approved