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Coefficients of power series for (1 - (1-8*x)^(1/4))/2.
5

%I #32 Aug 10 2022 02:59:24

%S 1,3,14,77,462,2926,19228,129789,894102,6258714,44379972,318056466,

%T 2299792908,16755634044,122874649656,906200541213,6716545187814,

%U 50000947509282,373691291911476,2802684689336070,21086865757861860,159109987082048580,1203701641403324040

%N Coefficients of power series for (1 - (1-8*x)^(1/4))/2.

%H Michael De Vlieger, <a href="/A048779/b048779.txt">Table of n, a(n) for n = 1..1112</a>

%H Elżbieta Liszewska, Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.

%F a(n) = 2^(n-1)*3*7*11*...*(4n-5)/n! = 2*a(n-1)*(32*a(n-2) + a(n-1))/(18*a(n-2) -a(n-1)).

%F a(n) = -A004984(n)/2.

%F D-finite with recurrence n*a(n) + 2*(5-4*n)*a(n-1) = 0. - _R. J. Mathar_, Oct 29 2012

%F G.f. A(x) =: y satisfies x = y * (1 - y) * (1 - 2*y + 2*y^2). - _Michael Somos_, Jan 17 2014

%F 0 = a(n) * (64*a(n+1) - 18*a(n+2)) + a(n+1) * (2*a(n+1) + a(n+2)) unless n=0. - _Michael Somos_, Jan 17 2014

%F From _Karol A. Penson_, Dec 19 2015: (Start)

%F a(n) = 8^(n-1)*binomial(n-5/4, -1/4)/n.

%F E.g.f.: is the hypergeometric function of type 1F1, in Maple notation hypergeom([3/4], [2], 8*x).

%F Representation as n-th moment of a positive function on (0, 8): a(n) = int(x^n*((2^(1/4)/(2*Pi*x^(1/4))*(1-x/8)^(1/4))), x=0..8), n=0,1,... . This function is the solution of the Hausdorff moment problem on (0, 8) with moments equal to a(n). As a consequence this representation is unique. (End)

%e G.f.: x + 3*x^2 + 14*x^3 + 77*x^4 + 462*x^5 + 2926*x^6 + 19228*x^7 + ...

%t a[ n_]:= If[n<1, 0, (-1/2)Pochhammer[-1/4, n] 8^n/n!] (* _Michael Somos_, Jan 17 2014 *)

%t a[ n_]:= SeriesCoefficient[(1 -(1-8x)^(1/4))/2, {x,0,n}] (* _Michael Somos_, Jan 17 2014 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( (1 - (1 - 8*x + x * O(x^n))^(1/4)) / 2, n))} /* _Michael Somos_, Jan 17 2014 */

%o (Magma) [Round(8^(n-1)*Gamma(n-1/4)/(Gamma(3/4)*Gamma(n+1))): n in [1..40]]; // _G. C. Greubel_, Aug 09 2022

%o (SageMath) [8^(n-1)*binomial(n-5/4,-1/4)/n for n in (1..40)] # _G. C. Greubel_, Aug 09 2022

%Y Related to Catalan numbers (A000108).

%Y Cf. A004984.

%K nonn

%O 1,2

%A _Jan Kristian Haugland_