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URL: https://oeis.org/A298799

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A298799
Expansion of (1-27*x)^(-1/9).
4
1, 3, 45, 855, 17955, 398601, 9167823, 216098685, 5186368440, 126201632040, 3104560148184, 77049538223112, 1926238455577800, 48452305767226200, 1225151160114148200, 31118839466899364280, 793530406405933789140, 20305042752151835192700, 521162763971897103279300
OFFSET
0,2
COMMENTS
Conjecture: a(p*n) == a(n) (mod p^2) for prime p == 1 (mod 9) and all positive integers n except those n of the form n = m*p + k for 0 <= m <= (p-1)/9 and 1 <= k <= (p-1)/9. Cf. A034171, A004981 and A004982. - Peter Bala, Dec 23 2019
LINKS
FORMULA
a(n) = 3^n/n! * Product_{k=0..n-1} (9*k + 1) for n > 0.
a(n) ~ 3^(3*n) / (Gamma(1/9) * n^(8/9)). - Vaclav Kotesovec, Jun 23 2018
From Peter Luschny, Dec 26 2019: (Start)
a(n) = (-27)^n*binomial(-1/9, n).
a(n) = n! * [x^n] hypergeom([1/9], [1], 27*x). (End)
D-finite with recurrence: n*a(n) + 3*(-9*n+8)*a(n-1) = 0. - R. J. Mathar, Jan 20 2020
MAPLE
seq(coeff(series((1-27*x)^(-1/9), x, n+1), x, n), n=0..20); # Muniru A Asiru, Jun 23 2018
# Alternative:
A298799 := n -> (-27)^n*binomial(-1/9, n):
seq(A298799(n), n=0..17); # Peter Luschny, Dec 26 2019
MATHEMATICA
a[n_] := (-27)^n * Binomial[-1/9, n]; Array[a, 20, 0] (* Amiram Eldar, Dec 22 2025 *)
PROG
(PARI) N=20; x='x+O('x^N); Vec((1-27*x)^(-1/9))
(GAP) List([0..20], n->(3^n/Factorial(n))*Product([0..n-1], k->9*k+1)); # Muniru A Asiru, Jun 23 2018
CROSSREFS
(1-b*x)^(-1/A003557(b)): A000984 (b=4), A004981 (b=8), A004987 (b=9), A098658 (b=12), A224881 (b=16), A034688 (b=25), this sequence (b=27), A004993 (b=36), A034835 (b=49).
Sequence in context: A266698 A132303 A386876 * A202437 A008931 A036278
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jun 22 2018
STATUS
approved