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A013698
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a(n) = binomial(3*n+2, n-1).
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15
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1, 8, 55, 364, 2380, 15504, 100947, 657800, 4292145, 28048800, 183579396, 1203322288, 7898654920, 51915526432, 341643774795, 2250829575120, 14844575908435, 97997533741800, 647520696018735, 4282083008118300
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OFFSET
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1,2
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COMMENTS
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Degree of variety K_{2,n}^1. Also number of double-rises (or odd-level peaks) in all generalized {(1,2),(1,-1)}-Dyck paths of length 3(n+1).
Number of dissections of a convex (2n+2)-gon by n-2 noncrossing diagonals into (2j+2)-gons, 1<=j<=n-1.
a(n) is the number of lattice paths from (0,0) to (3n+1,n-1) avoiding two consecutive up-steps. - Shanzhen Gao, Apr 20 2010
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LINKS
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FORMULA
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G.f.: g/((g-1)^3*(3*g-1)) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 09 2011
D-finite with recurrence 2*(2*n+3)*(n+1)*a(n) -n*(67*n+34)*a(n-1) +30*(3*n-1)*(3*n-2)*a(n-2)=0. - R. J. Mathar, Feb 05 2013
a(n+1) = (3*n+5)*(3*n+4)*(3*n+3)*a(n)/((2*n+5)*(2*n+4)*n). - Robert Israel, Aug 09 2015
With offset 0, the o.g.f. equals f(x)*g(x)^5, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k,n). Cf. A045721 (k = 1), A025174 (k = 2), A004319 (k = 3), A236194 (k = 4), A165817 (k = -1), A117671 (k = -2). - Peter Bala, Nov 04 2015
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MAPLE
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MATHEMATICA
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PROG
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(PARI) first(m)=vector(m, n, binomial(3*n+2, n-1)); /* Anders Hellström, Aug 09 2015 */
(Sage) [binomial(3*n+2, n-1) for n in (1..25)] # G. C. Greubel, Mar 21 2019
(GAP) List([1..25], n-> Binomial(3*n+2, n-1)) # G. C. Greubel, Mar 21 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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