A224881 -id:A224881 - cp's OEIS frontend

A298799 Expansion of (1-27*x)^(-1/9).

1, 3, 45, 855, 17955, 398601, 9167823, 216098685, 5186368440, 126201632040, 3104560148184, 77049538223112, 1926238455577800, 48452305767226200, 1225151160114148200, 31118839466899364280, 793530406405933789140, 20305042752151835192700

Offset: 0

Seiichi Manyama, Jun 22 2018

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 0..700

(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).

Cf. A305991, A034171, A004981, A004982.

List([0..20],n->(3^n/Factorial(n))*Product([0..n-1],k->9*k+1)); # Muniru A Asiru, Jun 23 2018

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

N=20; x='x+O('x^N); Vec((1-27*x)^(-1/9))

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

A376836 Expansion of 1/((1-x)^8 - 16*x^8)^(1/8).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 19, 91, 331, 991, 2575, 6007, 12871, 25759, 48927, 90271, 168607, 339151, 773263, 1982575, 5393455, 14709625, 38941801, 98802081, 239718961, 557984701, 1254360781, 2750444101, 5961934261, 12986219371, 28897996843, 66459189259, 158256566091

Offset: 0

Seiichi Manyama, Oct 06 2024

nonn

Cf. A002426, A376835.

Cf. A224881.

a[n_]:=Sum[(-16)^k * Binomial[-1/8,k] * Binomial[n,n-8*k],{k,0,Floor[n/8]}]; Array[a,36,0] (* Stefano Spezia, Oct 06 2024 *)

my(N=40, x='x+O('x^N)); Vec(1/((1-x)^8-16*x^8)^(1/8))

a(n) = Sum_{k=0..floor(n/8)} (-16)^k * binomial(-1/8,k) * binomial(n,n-8*k).

A224882 G.f.: 1/(1 - 32*x)^(1/16).

Original entry on oeis.org

1, 2, 34, 748, 18326, 476476, 12864852, 356540184, 10072260198, 288738125676, 8373405644604, 245112419778408, 7230816383463036, 214699624924363992, 6410317372741724904, 192309521182251747120, 5793324325615333881990, 175162864903898918549580

Offset: 0

Paul D. Hanna, Jul 23 2013

nonn

			G.f.: A(x) = 1 + 2*x + 34*x^2 + 748*x^3 + 18326*x^4 + 476476*x^5 +...
where
A(x)^16 = 1 + 32*x + 1024*x^2 + 32768*x^3 + 1048576*x^4 +...+ 32^n*x^n +...
Also,
A(x)^8 = 1 + 16*x + 384*x^2 + 10240*x^3 + 286720*x^4 +...+ 8^n*A000984(n)*x^n +...
A(x)^4 = 1 + 8*x + 160*x^2 + 3840*x^3 + 99840*x^4 +...+ 4^n*A004981(n)*x^n +...
A(x)^2 = 1 + 4*x + 72*x^2 + 1632*x^3 + 40800*x^4 +...+ 2^n*A224881(n)*x^n +...

Cf. A224881, A000984, A004981.

CoefficientList[Series[1/(1-32*x)^(1/16), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 24 2013 *)

{a(n)=polcoeff(1/(1-32*x +x*O(x^n))^(1/16),n)}
for(n=0,30,print1(a(n),", "))

{a(n)=(2^n/n!)*prod(k=0, n-1, 16*k + 1)}
for(n=0, 30, print1(a(n), ", "))

a(n) = (2^n/n!) * Product_{k=0..n-1} (16*k + 1).

a(n) ~ 32^n/(GAMMA(1/16)*n^(15/16)). - Vaclav Kotesovec, Jul 24 2013

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A298799 Expansion of (1-27*x)^(-1/9).

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A376836 Expansion of 1/((1-x)^8 - 16*x^8)^(1/8).

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A224882 G.f.: 1/(1 - 32*x)^(1/16).

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