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

%I A298799 #43 Jan 30 2020 21:29:18
%S A298799 1,3,45,855,17955,398601,9167823,216098685,5186368440,126201632040,
%T A298799 3104560148184,77049538223112,1926238455577800,48452305767226200,
%U A298799 1225151160114148200,31118839466899364280,793530406405933789140,20305042752151835192700
%N A298799 Expansion of (1-27*x)^(-1/9).
%C A298799 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 A298799 Seiichi Manyama, <a href="/A298799/b298799.txt">Table of n, a(n) for n = 0..700</a>
%F A298799 a(n) = 3^n/n! * Product_{k=0..n-1} (9*k + 1) for n > 0.
%F A298799 a(n) ~ 3^(3*n) / (Gamma(1/9) * n^(8/9)). - _Vaclav Kotesovec_, Jun 23 2018
%F A298799 From _Peter Luschny_, Dec 26 2019: (Start)
%F A298799 a(n) = (-27)^n*binomial(-1/9, n).
%F A298799 a(n) = n! * [x^n] hypergeom([1/9], [1], 27*x). (End)
%F A298799 D-finite with recurrence: n*a(n) +3*(-9*n+8)*a(n-1)=0. - _R. J. Mathar_, Jan 20 2020
%p A298799 seq(coeff(series((1-27*x)^(-1/9), x, n+1), x, n), n=0..20); # _Muniru A Asiru_, Jun 23 2018
%p A298799 # Alternative:
%p A298799 A298799 := n -> (-27)^n*binomial(-1/9, n):
%p A298799 seq(A298799(n), n=0..17); # _Peter Luschny_, Dec 26 2019
%o A298799 (PARI) N=20; x='x+O('x^N); Vec((1-27*x)^(-1/9))
%o A298799 (GAP) List([0..20],n->(3^n/Factorial(n))*Product([0..n-1],k->9*k+1)); # _Muniru A Asiru_, Jun 23 2018
%Y A298799 (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 A298799 Cf. A305991, A034171, A004981, A004982.
%K A298799 nonn
%O A298799 0,2
%A A298799 _Seiichi Manyama_, Jun 22 2018