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A025757
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4th order Vatalan numbers (generalization of Catalan numbers).
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2
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1, 1, 7, 69, 783, 9597, 123495, 1643397, 22413183, 311466829, 4392857431, 62702224213, 903886452975, 13138698859677, 192337495360071, 2832859169364261, 41946319269028191, 624009420903043821
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OFFSET
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0,3
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
T. M. Richardson, The Super Patalan Numbers, arXiv preprint arXiv:1410.5880, 2014
T. M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015) # 15.3.3.
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FORMULA
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G.f.: 4 / (3+(1-16*x)^(1/4)).
a(n) = sum(m=1..n-1, m/n*4^(n-m) * sum(k=1..n-m, binomial(n+k-1,n-1) * sum(j=0..k, binomial(j,n-m-3*k+2*j) * 4^(j-k) * binomial(k,j) * 3^(-n+m+3*k-j) * 2^(n-m-3*k+j) * (-1)^(n-m-3*k+2*j))))+1. [From Vladimir Kruchinin, Feb 08 2011]
Conjecture: 5*n*(n-1)*(n-2)*a(n) -(239*n-600)*(n-1)*(n-2)*a(n-1) +24*(n-2)*(158*n^2-953*n+1445)*a(n-2) +16*(-1232*n^3+13056*n^2-45949*n+53730)*a(n-3) -128*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4)=0. - R. J. Mathar, Jul 28 2014
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MATHEMATICA
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Table[SeriesCoefficient[4/(3 + (1 - 16*x)^(1/4)), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 29 2012 *)
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CROSSREFS
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a(n), n >= 1, = row sums of triangle A049213.
Sequence in context: A122010 A180911 A084774 * A243668 A265033 A226270
Adjacent sequences: A025754 A025755 A025756 * A025758 A025759 A025760
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KEYWORD
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nonn
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AUTHOR
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Olivier Gérard
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STATUS
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approved
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