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A025756
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3rd-order Vatalan numbers (generalization of Catalan numbers).
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2
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1, 1, 4, 22, 139, 949, 6808, 50548, 384916, 2988418, 23559826, 188061592, 1516680130, 12337999870, 101111413540, 833914857316, 6916004156083, 57638242134229, 482444724374734, 4053815358183454, 34181335453533439
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f.: 3 / (2+(1-9*x)^(1/3)).
a(n) = Sum_{m=1..n-1} (m/n) * Sum_{k=1..n-m} binomial(k,n-m-k) * 3^k * (-1)^(n-m-k) * binomial(n+k-1,n-1) + 1. - Vladimir Kruchinin, Feb 08 2011
Conjecture: n*(n-1)*a(n) -(n-1)*(19*n-36)*a(n-1) +9*(11*n^2-51*n+60)*a(n-2) -9*(3*n-7)*(3*n-8)*a(n-3) = 0. - R. J. Mathar, Nov 14 2011
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MAPLE
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coeftayl( 3/(2+(1-9*x)^(1/3)), x=0, n);
end proc:
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MATHEMATICA
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Table[SeriesCoefficient[3/(2+(1-9*x)^(1/3)), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 08 2012 *)
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PROG
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(Maxima) a[0]:1$ a[n]:=(1/n)*((9*n-6)*a[n-1]-2*sum(a[k]*a[n-1-k], k, 0, n-1))$ makelist(a[n], n, 0, 1000); /* Tani Akinari, Aug 02 2014 */
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CROSSREFS
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Row sums of triangle A048966, n > 0.
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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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