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A004117 Numerators of expansion of (1-x)^(-1/3). 4

%I

%S 1,1,2,14,35,91,728,1976,5434,135850,380380,1071980,9111830,25933670,

%T 74096200,637227320,1832028545,5280552865,137294374490,397431084050,

%U 1152550143745,10043651252635,29217894553120,85112997176480

%N Numerators of expansion of (1-x)^(-1/3).

%C For n>=1 a(n) is also the numerator of Beta(n+1/3,2/3)*sqrt(27)/(2*Pi). [Groux Roland, May 17 2011]

%H Vincenzo Librandi, <a href="/A004117/b004117.txt">Table of n, a(n) for n = 0..1000</a>

%F (1/n!) * prod(k=0, n-1, 3k+1) * 3^A054861(n). - _Ralf Stephan_, Mar 13 2004

%F Numerators in (1-3t)^(-1/3) = 1 + t + 2*t^2 + (14/3)*t^3 + (35/3)*t^4 + (91/3)*t^5 + (728/9)*t^6 + (1976/9)*t^7 + (5434/9)*t^8 + ... = 1 + t + 4*t^2/2! + 28*t^3/3! + 280*t^4/4! + 3640*t^5/5! + 58240*t^6/6! + ... = e.g.f. for triple factorials A007559 (cf. A094638). - _Tom Copeland_, Dec 04 2013

%t Table[Numerator[Binomial[-1/3,n] (-1)^n],{n,0,40}] (* _Vincenzo Librandi_, Jun 13 2012 *)

%o (PARI) a(n)=prod(k=1,n,3*k-2)/n!*3^sum(k=1,n,valuation(k,3))

%Y Cf. A004128.

%Y Cf. A004987, A007559, A047657, A054861, A034689, A053101, A072888.

%K nonn

%O 0,3

%A _N. J. A. Sloane_.

%E Fixed typo in formula. - _Pontus von Brömssen_, Nov 25 2008

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Last modified October 21 23:02 EDT 2018. Contains 316431 sequences. (Running on oeis4.)