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A298799
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Expansion of (1-27*x)^(-1/9).
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4
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1, 3, 45, 855, 17955, 398601, 9167823, 216098685, 5186368440, 126201632040, 3104560148184, 77049538223112, 1926238455577800, 48452305767226200, 1225151160114148200, 31118839466899364280, 793530406405933789140, 20305042752151835192700
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) = 3^n/n! * Product_{k=0..n-1} (9*k + 1) for n > 0.
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
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MAPLE
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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):
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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