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Expansion of (1-27*x)^(-1/9).
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%I #43 Jan 30 2020 21:29:18

%S 1,3,45,855,17955,398601,9167823,216098685,5186368440,126201632040,

%T 3104560148184,77049538223112,1926238455577800,48452305767226200,

%U 1225151160114148200,31118839466899364280,793530406405933789140,20305042752151835192700

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

%C 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

%H Seiichi Manyama, <a href="/A298799/b298799.txt">Table of n, a(n) for n = 0..700</a>

%F a(n) = 3^n/n! * Product_{k=0..n-1} (9*k + 1) for n > 0.

%F a(n) ~ 3^(3*n) / (Gamma(1/9) * n^(8/9)). - _Vaclav Kotesovec_, Jun 23 2018

%F From _Peter Luschny_, Dec 26 2019: (Start)

%F a(n) = (-27)^n*binomial(-1/9, n).

%F a(n) = n! * [x^n] hypergeom([1/9], [1], 27*x). (End)

%F D-finite with recurrence: n*a(n) +3*(-9*n+8)*a(n-1)=0. - _R. J. Mathar_, Jan 20 2020

%p seq(coeff(series((1-27*x)^(-1/9), x, n+1), x, n), n=0..20); # _Muniru A Asiru_, Jun 23 2018

%p # Alternative:

%p A298799 := n -> (-27)^n*binomial(-1/9, n):

%p seq(A298799(n), n=0..17); # _Peter Luschny_, Dec 26 2019

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

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

%Y (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).

%Y Cf. A305991, A034171, A004981, A004982.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jun 22 2018