A036910 a(n) = (binomial(4*n, 2*n) + binomial(2*n, n)^2)/2.

1, 5, 53, 662, 8885, 124130, 1778966, 25947612, 383358645, 5719519850, 85990654178, 1300866635172, 19780031677718, 302045506654052, 4629016098160220, 71163013287905912, 1096960888092571317, 16949379732631632570, 262435310495071434602

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972, Eq 3.68, page 30.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 6.

Crossrefs

Cf. A000984, A001448, A157531.

Programs

Magma

[(Binomial(4*n, 2*n) + Binomial(2*n, n)^2)/2: n in [0..40]]; // G. C. Greubel, Dec 09 2021

Mathematica

B[n_] := Binomial[2*n, n]/2; Table[B[2*n] + 2*B[n]^2, {n, 0, 40}] (* G. C. Greubel, Dec 09 2021 *)

PARI

a(n) = (binomial(4*n,2*n)+binomial(2*n,n)^2)/2; \\ Michel Marcus, Dec 09 2021

Sage

[(binomial(4*n, 2*n) + binomial(2*n, n)^2)/2 for n in (0..40)] # G. C. Greubel, Dec 09 2021

Formula

a(n) = Sum_{k=0..n} binomial(2n, k)^2. - Paul Barry, May 15 2003

From G. C. Greubel, Dec 09 2021: (Start)

a(n) = A000984(2*n) + A000984(n)^2.

a(n) = A001448(n) + A000984(n)^2.

a(n) = A157531(2n, n)/2. - corrected Jan 12 2023

G.f.: sqrt(1 + sqrt(1 - 16*x))/(2*sqrt(2)*sqrt(1 - 16*x)) + (1/Pi)*EllipticK[16*x]. (End)

D-finite with recurrence n^2*(2*n-1)*(n-1)*a(n) -2*(n-1) *(2*n^3+109*n^2-241*n+132) *a(n-1) +12*(-112*n^4+1008*n^3-3011*n^2+3702*n-1600) *a(n-2) +16*(832*n^4-7008*n^3+20744*n^2-24228*n+7905) *a(n-3) +256*(4*n-13) *(4*n-15)*(-7+2*n)^2*a(n-4)=0. - R. J. Mathar, Jan 12 2023