A277247 a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)^2.
1, 1, 5, 10, 53, 126, 662, 1716, 8885, 24310, 124130, 352716, 1778966, 5200300, 25947612, 77558760, 383358645, 1166803110, 5719519850, 17672631900, 85990654178, 269128937220, 1300866635172, 4116715363800, 19780031677718, 63205303218876, 302045506654052
Offset: 0
Programs
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Maple
A277247 := proc(n) add(binomial(n,k)^2,k=0..floor(n/2)) ; end proc: seq(A277247(n),n=0..50) ; # R. J. Mathar, Jan 11 2024
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Mathematica
Table[(Binomial[2 n, n] + (Binomial[n, n/2] Cos[Pi n/2])^2)/2, {n, 0, 30}] CoefficientList[Series[(1/Sqrt[1-4x]+(2EllipticK[16 x^2])/Pi)/2, {x, 0, 20}], x] (* Benedict W. J. Irwin, Oct 19 2016 *)
Formula
a(n) = (binomial(2*n, n) + (binomial(n, n/2)*cos(Pi*n/2))^2)/2.
D-finite with recurrence: 2*(2*n+1)*(4*n^2+15*n+13)*(16*(n+1)^2*a(n) - (n+2)^2*a(n+2)) = (n+2)*(4*n^2+7*n+2)*(16*(n+2)^2*a(n+1) - (n+3)^2*a(n+3)).
G.f.: (1/sqrt(1 - 4*x) + 2*K(4*x)/Pi)/2, where K is the complete elliptic integral of the first kind with modulus 4*x. - Benedict W. J. Irwin, Oct 19 2016
D-finite with recurrence n^2*(n-1)*a(n) -2*(3*n-4)*(n-1)^2*a(n-1) +4*(-19*n^2+64*n-56)*a(n-2) +16*(4*n^3-11*n^2-16*n+49)*a(n-3) -64*(4*n-15)*(n-3)^2*a(n-4) +256*(2*n-9)*(n-4)^2*a(n-5)=0. - R. J. Mathar, Jan 11 2024
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