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Numerators in expansion of (1-x)^{-1/4}.
6

%I #31 May 09 2014 02:05:10

%S 1,1,5,15,195,663,4641,16575,480675,1762475,13042315,48612265,

%T 729183975,2748462675,20809788825,79077197535,4823709049635,

%U 18443593425075,141400882925575,543277076503525,8366466978154285,32270658344309385

%N Numerators in expansion of (1-x)^{-1/4}.

%C Numerators in expansion of sqrt(1/sqrt(1-4x)). - _Paul Barry_, Jul 12 2005

%C Denominators are in A088802. - _Michael Somos_, Aug 23 2007

%H Vincenzo Librandi, <a href="/A004130/b004130.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = prod(k=1, n, (4k-3)/k * 2^A007814(k)), proved by _Mitch Harris_, following a conjecture by _Ralf Stephan_.

%F a(n) = 2^(e_2((2n)!)-n)/n! Product[4k+1,{k,0,n-1}], where e_2((2n)!) is the highest power of 2 that divides (2n)! (sequence A005187). - _Emanuele Munarini_, Jan 25 2011

%F Numerators in (1-4t)^(-1/4) = 1 + t + (5/2)t^2 + (15/2)t^3 + (195/8)t^4 + (663/8)t^5 + (4641/16)t^6 + (16575/16)t^7 + ... = 1 + t + 5*t^2/2! + 45*t^3/3! + 585*t^4/4! + ... = e.g.f. for the quartic factorials A007696 (cf. A094638). - _Tom Copeland_, Dec 04 2013

%t Table[Numerator[Binomial[-1/4, n] (-1)^n], {n, 0, 20}]

%o (PARI) {a(n) = if( n<0, 0, numerator( polcoeff( (1 - x +x*O(x^n))^(-1/4), n ) ) ) } /* _Michael Somos_, Aug 23 2007 */

%Y Cf. A004134, A004981, A034255, A034385, A048882, A007696, A000265, A049606.

%K nonn,frac

%O 0,3

%A _N. J. A. Sloane_