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A052203
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a(n) = (4n+1)*binomial(4n,n)/(3n+1).
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16
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1, 5, 36, 286, 2380, 20349, 177100, 1560780, 13884156, 124403620, 1121099408, 10150595910, 92263734836, 841392966470, 7694644696200, 70539987842520, 648045936942300, 5964720367660956, 54991682779774384, 507749884105448600, 4694436188839116720
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OFFSET
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0,2
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COMMENTS
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a(n) is the number of paths from (0,0) to (4n,n), taking north and east steps while avoiding exactly 2 consecutive north steps. - Shanzhen Gao, Apr 15 2010
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LINKS
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FORMULA
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a(n) = C(4n+1, n); a(n) is asymptotic to c/sqrt(n)*(256/27)^n with c=0.614... - Benoit Cloitre, Jan 27 2003 [c = 2^(5/2)/(3^(3/2)*sqrt(Pi)) = 0.61421182128... - Vaclav Kotesovec, Feb 14 2019]
G.f.: hypergeom([1/2, 3/4, 5/4], [2/3, 4/3], (256/27)*x). - Robert Israel, Aug 07 2014
D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n) - 8*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Nov 26 2012
The o.g.f. equals f(x)*g(x), where f(x) is the o.g.f. for A005810 and g(x) is the o.g.f. for A002293.
More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(4*n + k,n). Cf. A262977 (k = -1), A005810 (k = 0), A257633 (k = 2), A224274 (k = 3) and A004331 (k = 4). (End)
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MATHEMATICA
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PROG
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(Haskell)
(PARI) vector(30, n, n--; (4*n+1)*binomial(4*n, n)/(3*n+1)) \\ Altug Alkan, Nov 05 2015
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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