A052203 a(n) = (4n+1)*binomial(4n,n)/(3n+1).

1, 5, 36, 286, 2380, 20349, 177100, 1560780, 13884156, 124403620, 1121099408, 10150595910, 92263734836, 841392966470, 7694644696200, 70539987842520, 648045936942300, 5964720367660956, 54991682779774384, 507749884105448600, 4694436188839116720

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 * A371775 A332624 A027331

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