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A025748
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3rd-order Patalan numbers (generalization of Catalan numbers).
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14
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1, 1, 3, 15, 90, 594, 4158, 30294, 227205, 1741905, 13586859, 107459703, 859677624, 6943550040, 56540336040, 463630755528, 3824953733106, 31724616256938, 264371802141150, 2212374554760150, 18583946259985260, 156636118477018620, 1324287183487521060
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OFFSET
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0,3
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COMMENTS
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G.f. (with a(0)=0) is series reversion of x - 3*x^2 + 3*x^3.
The Hankel transform of a(n) is A005130(n) * 3^binomial(n,2).
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LINKS
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FORMULA
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G.f.: (4 - (1-9*x)^(1/3))/3.
a(n) = 3^(n-1) * 2 * A034000(n-1)/n!, n >= 2.
a(n) = 3 * A034164(n-2), n >= 2. (End)
D-finite with recurrence n*a(n) +3*(4-3*n)*a(n-1) = 0, n >= 2. - R. J. Mathar, Oct 29 2012
For n>0, a(n) = 9^(n-1) * Gamma(n-1/3) / (n * Gamma(2/3) * Gamma(n)). - Vaclav Kotesovec, Feb 09 2014
For n > 0, a(n) = 3^(2*n-1)*(-1)^(n+1)*binomial(1/3, n). - Peter Bala, Mar 01 2022
Sum_{n>=0} 1/a(n) = 37/16 + 3*sqrt(3)*Pi/64 - 9*log(3)/64. - Amiram Eldar, Dec 02 2022
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MAPLE
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local x;
coeftayl(4-(1-9*x)^(1/3), x=0, n) ;
%/3 ;
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MATHEMATICA
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CoefficientList[Series[(4-Power[1-9x, (3)^-1])/3, {x, 0, 25}], x] (* Harvey P. Dale, Nov 14 2011 *)
Flatten[{1, Table[FullSimplify[9^(n-1) * Gamma[n-1/3] / (n * Gamma[2/3] * Gamma[n])], {n, 1, 25}]}] (* Vaclav Kotesovec, Feb 09 2014 *)
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PROG
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(PARI) a(n)=if(n<1, n==0, polcoeff(serreverse(x-3*x^2+3*x^3+x*O(x^n)), n))
(Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (4 - (1-9*x)^(1/3))/3 )); // G. C. Greubel, Sep 17 2019
(Sage)
P.<x> = PowerSeriesRing(QQ, prec)
return P((4 - (1-9*x)^(1/3))/3).list()
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CROSSREFS
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Apart from the initial 1, identical to A097188.
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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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