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 A025748 3rd order Patalan numbers (generalization of Catalan numbers). 14
 1, 1, 3, 15, 90, 594, 4158, 30294, 227205, 1741905, 13586859, 107459703, 859677624, 6943550040, 56540336040, 463630755528, 3824953733106, 31724616256938, 264371802141150, 2212374554760150, 18583946259985260 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 I. M. Gessel, G. Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005, eq. (5.1). W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4. Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019. T. M. Richardson, The Super Patalan Numbers, arXiv preprint arXiv:1410.5880 [math.CO], 2014 and J. Int. Seq. 18 (2015) # 15.3.3. FORMULA From Wolfdieter Lang: (Start) 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) 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 MAPLE A025748 :=proc(n)         local x;         coeftayl(4-(1-9*x)^(1/3), x=0, n) ;         %/3 ; end proc: # R. J. Mathar, Nov 01 2012 MATHEMATICA 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 *) PROG (PARI) a(n)=if(n<1, n==0, polcoeff(serreverse(x-3*x^2+3*x^3+x*O(x^n)), n)) (MAGMA) R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (4 - (1-9*x)^(1/3))/3 )); // G. C. Greubel, Sep 17 2019 (Sage) def A025748_list(prec):     P. = PowerSeriesRing(QQ, prec)     return P((4 - (1-9*x)^(1/3))/3).list() A025748_list(25) # G. C. Greubel, Sep 17 2019 CROSSREFS Apart from the initial 1, identical to A097188. Cf. A005130, A034000, A034164. Sequence in context: A205576 A173695 A255688 * A097188 A271930 A201953 Adjacent sequences:  A025745 A025746 A025747 * A025749 A025750 A025751 KEYWORD nonn AUTHOR STATUS approved

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Last modified December 2 23:36 EST 2020. Contains 338898 sequences. (Running on oeis4.)