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
a(n) = Sum_{k = 0..n} binomial(2*n+k+1, k)*binomial(2*n-k-1, n-k). - Peter Bala, Sep 17 2024
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
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, Jan 28 2000
EXTENSIONS
More terms from James A. Sellers, Jan 31 2000
STATUS
approved