A241530 a(n) = binomial(n,floor(n/2))*binomial(n+1,floor(n/2+1/2))*(1+floor(n/2))/(1+2*floor(n/2)).

1, 2, 4, 12, 36, 120, 400, 1400, 4900, 17640, 63504, 232848, 853776, 3171168, 11778624, 44169840, 165636900, 625739400, 2363904400, 8982836720, 34134779536, 130332794592, 497634306624, 1907598175392, 7312459672336, 28124844893600, 108172480360000

Offset: 0

Author

Peter Luschny, Apr 25 2014

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000894, A002894, A005566.

Row n=3 of A275784.

Programs

Maple

A241530 := n -> binomial(n,iquo(n,2))*binomial(n+1,iquo(n+1,2))
*(1+iquo(n,2))/(1+2*iquo(n,2)); seq(A241530(n), n=0..26);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 2^n,
     ((8*n-4)*a(n-1)+16*(n-1)*(n-2)*a(n-2))/(n*(n+1)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 10 2016

Mathematica

CoefficientList[Series[(-EllipticE[16 x^2] + (1 + 4 x) EllipticK[16 x^2])/(2Pi x), {x, 0, 20}], x] (* Benedict W. J. Irwin, Aug 15 2016 *)
Table[Binomial[n, #] Binomial[n + 1, Floor[(n + 1)/2]] (1 + #)/(1 + 2 #) &@ Floor[n/2], {n, 0, 26}] (* Michael De Vlieger, Aug 15 2016 *)

Formula

a(n) = ((8*n-4)*a(n-1)+16*(n-1)*(n-2)*a(n-2))/(n*(n+1)) for n>=2, a(n) = 2^n for n<2. - Alois P. Heinz, Apr 25 2014

G.f.: ((1+4*x)*K(4*x) - E(4*x))/(2*Pi*x), where K and E are the complete elliptic integrals of the first and second kind, respectively, with modulus k = 4*x. - Benedict W. J. Irwin, Aug 15 2016

From Wolfdieter Lang, Sep 06 2016 (Start):

The preceding g.f. can be rewritten as ((1+4*x)*F(1/2,1/2;1;(4*x)^2) -

F(-1/2,1/2;1;(4*x)^2))/(4*x), where F is the hypergyometric function F(a,b;c;z).

This leads to the bisection a(2*k) = ((2*k)!)^2/k!^4 = A002894(k) and a(2*k+1) = 2*(2*k)!*(2*k+1)!/((k+1)*k!^4) = 2*A000894(k), for k >= 0.

(End)