A002458 a(n) = binomial(4*n+1, 2*n).

1, 10, 126, 1716, 24310, 352716, 5200300, 77558760, 1166803110, 17672631900, 269128937220, 4116715363800, 63205303218876, 973469712824056, 15033633249770520, 232714176627630544, 3609714217008132870, 56093138908331422716, 873065282167813104916

Offset

0,2

References

The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1982, (3.109), page 35.

Links

T. D. Noe, Table of n, a(n) for n = 0..100

Formula

a(n) = Sum_{k=0..n} 4^k * binomial( n + k, n) * binomial( 2*n - 2*k, n - k). - Michael Somos, Feb 25 2012

a(n) = A001700(2*n) = (n+1)*A000108(2*n+1).

G.f.: (4 - (1+4*y)*c(y) - (1-4*y)*c(-y))/(2*(1 - (4*y)^2)) with y^2 = x, c(y) = g.f. for A000108 (Catalan). - Wolfdieter Lang, Dec 13 2001

a(n) ~ 2^(1/2)*Pi^(-1/2)*n^(-1/2)*2^(4*n)*{1 - 5/16*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Jun 11 2002

a(n) = A024492(n)*(n+1). - R. J. Mathar, Aug 10 2015

G.f.: 2F1(3/4,5/4; 3/2; 16*x). - R. J. Mathar, Aug 10 2015

D-finite with recurrence n*(2*n + 1)*a(n) - 2*(4*n - 1)*(4*n + 1)*a(n-1) = 0. - R. J. Mathar, Aug 10 2015

From Peter Bala, Nov 04 2015: (Start)

a(n) = 4^n*binomial(2*n + 1/2, n).

O.g.f.: sqrt(c(4*x)/(1 - 16*x)) = sqrt(2/(1 - 16*x)/(1 + sqrt(1 - 16*x))), where

c(y) = g.f. for A000108 (Catalan). In general, c(x)^k/sqrt(1 - 4*x) is the o.g.f. for the sequence binomial(2*n + k, n). (End) [Edited by Petros Hadjicostas, May 25 2020]

From Ilya Gutkovskiy, Jan 17 2017: (Start)

E.g.f.: 2F2(3/4,5/4; 1,3/2; 16*x).

Sum_{n>=0} 1/a(n) = 3F2(1,1,3/2; 3/4,5/4; 1/16) = 1.108563435104316693... (End)

From Peter Bala, Mar 16 2018: (Start)

The right-hand side of the binomial coefficient identity Sum_{k = 0..n} 4^(n-k) * C(2*n+1, 2*k) * C(2*k, k) = a(n).

a(n) = 4^n*hypergeom([-n, -n-1/2], [1], 1). (End)

From Peter Bala, Mar 20 2023: (Start)

a(n) = Sum_{k = 0..n} binomial(2*n+1,k)^2.

a(n) = (1/2)*hypergeom([-1 - 2*n, -1 - 2*n], [1], 1). (End)

Example

1 + 10*x + 126*x^2 + 1716*x^3 + 24310*x^4 + 352716*x^5 + 5200300*x^6 + ...

Maple

A002458:=n->binomial(4*n+1, 2*n): seq(A002458(n), n=0..30); # Wesley Ivan Hurt, Jan 17 2017

Mathematica

Table[Binomial[4n+1, 2n], {n, 0, 30}] (* Harvey P. Dale, Apr 04 2011 *)
4^Range[0, 22] Simplify[ CoefficientList[ Series[ Sqrt[2]/(((Sqrt[1 - 4 x] + 1)^(1/2))*Sqrt[1 - 4 x]), {x, 0, 22}], x]] (* Robert G. Wilson v, Aug 08 2011 *)

Prog

(PARI) a(n) = binomial( 4*n + 1, 2*n)

Crossrefs

Cf. A000984, A001448, A001700, A024492, A100033, A187364, A187365.

Row sums of A067001.

Sequence in context: A323877 A079609 A101599 * A366234 A192600 A079241

Adjacent sequences: A002455 A002456 A002457 * A002459 A002460 A002461

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved