A259557 a(n) = binomial(4*n-1, 2*n).

1, 3, 35, 462, 6435, 92378, 1352078, 20058300, 300540195, 4537567650, 68923264410, 1052049481860, 16123801841550, 247959266474052, 3824345300380220, 59132290782430712, 916312070471295267, 14226520737620288370

Offset

0,2

Comments

Essentially the same as A100033.

Links

Table of n, a(n) for n=0..17.

V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.

Formula

G.f. A(x)=1+x*B(x)'/B(x), where B(x) is g.f. of A079489.

a(n) = A100033(n-1) for n>0.

D-finite with recurrence n*(2*n-1)*a(n) -2*(4*n-1)*(4*n-3)*a(n-1)=0. - R. J. Mathar, Jul 06 2015

a(n) = [x^(2*n)] 1/(1 - x)^(2*n). - Ilya Gutkovskiy, Oct 10 2017

From Peter Bala, Jun 11 2023: (Start)

a(n) = (1/2) * [x^n] ( (1 + x)^2/( 1 - x) )^(2*n) for n >= 1.

Right-hand side of the identity (1/2)*Sum_{k = 0..n} binomial(4*n,k)*binomial(3*n-k-1,n-k) = binomial(4*n-1,2*n) for n >= 1.

a(n) = [x^n] E(x)^n, where E(x) = exp( Sum_{k >= 1} A119259(k)*x^k/k ). (End)

a(n) = Sum_{k = 0..2*n} (-1)^k*binomial(-n, k)*binomial(-3*n-k, 2*n-k) = Sum_{k = 0..2*n} (-1)^k*binomial(n+k-1, k)*binomial(5*n-1, 2*n-k). - Peter Bala, Jun 08 2024

Mathematica

Table[Binomial[4 n - 1, 2 n], {n, 0, 30}] (* Vincenzo Librandi, Jul 01 2015 *)

Prog

(PARI) vector(20, n, n--; binomial(4*n-1, 2*n)) \\ Michel Marcus, Jul 01 2015
(Magma) [Binomial(4*n-1, 2*n): n in [0..20]]; // Vincenzo Librandi, Jul 01 2015

Crossrefs

Cf. A001700, A079489, A100033.

Sequence in context: A161495 A179135 A100033 * A184554 A046032 A304191

Adjacent sequences: A259554 A259555 A259556 * A259558 A259559 A259560

Keyword

nonn,easy

Author

Vladimir Kruchinin, Jun 30 2015

Status

approved