A037967 a(n) = (binomial(2*n, n)^2 + binomial(2*n, n))/2.

1, 3, 21, 210, 2485, 31878, 427350, 5891028, 82824885, 1181976510, 17067482146, 248817506028, 3656231188246, 54086245380300, 804670817838300, 12030722583033960, 180648817921816245, 2722858996178147310, 41179040361190612650, 624643836563467851900

Offset

0,2

References

The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972, Eq. (3.82), page 31.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..833 (first 101 terms from Chai Wah Wu)

Formula

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

n^2*(n-1)*(3*n-5)*a(n) - 2*(n-1)*(2*n-1)*(15*n^2-31*n+12)*a(n-1) + 8*(2*n-1)*(3*n-2)*(2*n-3)^2*a(n-2) = 0. - R. J. Mathar, Jul 26 2015

a(n) = A000217(A000984(n)). - Alois P. Heinz, Aug 10 2020

Maple

a:= n-> (h-> h*(h+1)/2)(binomial(2*n, n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 10 2020

Mathematica

Table[(Binomial[2n, n]^2 + Binomial[2n, n])/2, {n, 0, 45}] (* Vincenzo Librandi, Jun 02 2015 *)

Prog

(Python)
from gmpy2 import bincoef
def A037967(n):
    return bincoef(bincoef(2*n, n)+1, 2) # Chai Wah Wu, Jun 01 2015
(Magma) [(Binomial(2*n, n)^2+Binomial(2*n, n))/2: n in [0..30]]; // Vincenzo Librandi, Jun 02 2015
(PARI) a(n)=binomial(binomial(2*n, n)+1, 2) \\ Charles R Greathouse IV, Jun 02 2015
(SageMath) [binomial(1+(n+1)*catalan_number(n), 2) for n in (0..30)] # G. C. Greubel, Jun 19 2022

Crossrefs

Cf. A000108, A000217, A000984.

Sequence in context: A136223 A114469 A097690 * A123691 A087918 A088926

Adjacent sequences: A037964 A037965 A037966 * A037968 A037969 A037970

Keyword

nonn

Author

N. J. A. Sloane

Status

approved