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A048779
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Coefficients of power series for (1 - (1-8*x)^(1/4))/2.
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5
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1, 3, 14, 77, 462, 2926, 19228, 129789, 894102, 6258714, 44379972, 318056466, 2299792908, 16755634044, 122874649656, 906200541213, 6716545187814, 50000947509282, 373691291911476, 2802684689336070, 21086865757861860, 159109987082048580, 1203701641403324040
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OFFSET
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1,2
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LINKS
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FORMULA
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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)).
D-finite with recurrence n*a(n) + 2*(5-4*n)*a(n-1) = 0. - R. J. Mathar, Oct 29 2012
G.f. A(x) =: y satisfies x = y * (1 - y) * (1 - 2*y + 2*y^2). - Michael Somos, Jan 17 2014
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
a(n) = 8^(n-1)*binomial(n-5/4, -1/4)/n.
E.g.f.: is the hypergeometric function of type 1F1, in Maple notation hypergeom([3/4], [2], 8*x).
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)
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EXAMPLE
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G.f.: x + 3*x^2 + 14*x^3 + 77*x^4 + 462*x^5 + 2926*x^6 + 19228*x^7 + ...
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MATHEMATICA
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a[ n_]:= If[n<1, 0, (-1/2)Pochhammer[-1/4, n] 8^n/n!] (* Michael Somos, Jan 17 2014 *)
a[ n_]:= SeriesCoefficient[(1 -(1-8x)^(1/4))/2, {x, 0, n}] (* Michael Somos, Jan 17 2014 *)
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PROG
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(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 */
(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
(SageMath) [8^(n-1)*binomial(n-5/4, -1/4)/n for n in (1..40)] # G. C. Greubel, Aug 09 2022
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CROSSREFS
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Related to Catalan numbers (A000108).
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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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