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 A052203 a(n) = (4n+1)*binomial(4n,n)/(3n+1). 15
 1, 5, 36, 286, 2380, 20349, 177100, 1560780, 13884156, 124403620, 1121099408, 10150595910, 92263734836, 841392966470, 7694644696200, 70539987842520, 648045936942300, 5964720367660956, 54991682779774384, 507749884105448600, 4694436188839116720 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Central terms of the triangles in A122366 and A111418. - Reinhard Zumkeller, Aug 30 2006 and Mar 14 2014 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 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..1000 FORMULA 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.: g^2/(4-3*g) where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011 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 From Peter Bala, Nov 04 2015: (Start) 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) a(n) = [x^n] 1/(1 - x)^(3*n+2). - Ilya Gutkovskiy, Oct 03 2017 MATHEMATICA Table[Binomial[4 n + 1, n], {n, 0, 20}] (* Vincenzo Librandi, Aug 07 2014 *) PROG (Haskell) a052203 n = a122366 (2 * n) n  -- Reinhard Zumkeller, Mar 14 2014 (MAGMA) [Binomial(4*n+1, n): n in [0..20]]; // Vincenzo Librandi, Aug 07 2014 (PARI) vector(30, n, n--; (4*n+1)*binomial(4*n, n)/(3*n+1)) \\ Altug Alkan, Nov 05 2015 CROSSREFS Cf. A002293, A051944, A004331, A005810, A224274, A257633, A262977. Sequence in context: A201351 A253470 A188899 * A332624 A027331 A255489 Adjacent sequences:  A052200 A052201 A052202 * A052204 A052205 A052206 KEYWORD easy,nonn AUTHOR Barry E. Williams, Jan 28 2000 EXTENSIONS More terms from James A. Sellers, Jan 31 2000 STATUS approved

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Last modified January 25 06:09 EST 2021. Contains 340416 sequences. (Running on oeis4.)