This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A277247 #18 Jan 11 2024 07:08:53
%S A277247 1,1,5,10,53,126,662,1716,8885,24310,124130,352716,1778966,5200300,
%T A277247 25947612,77558760,383358645,1166803110,5719519850,17672631900,
%U A277247 85990654178,269128937220,1300866635172,4116715363800,19780031677718,63205303218876,302045506654052
%N A277247 a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)^2.
%C A277247 Interleaves A036910 and A002458.
%F A277247 a(n) = (binomial(2*n, n) + (binomial(n, n/2)*cos(Pi*n/2))^2)/2.
%F A277247 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)).
%F A277247 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
%F A277247 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
%p A277247 A277247 := proc(n)
%p A277247 add(binomial(n,k)^2,k=0..floor(n/2)) ;
%p A277247 end proc:
%p A277247 seq(A277247(n),n=0..50) ; # _R. J. Mathar_, Jan 11 2024
%t A277247 Table[(Binomial[2 n, n] + (Binomial[n, n/2] Cos[Pi n/2])^2)/2, {n, 0, 30}]
%t A277247 CoefficientList[Series[(1/Sqrt[1-4x]+(2EllipticK[16 x^2])/Pi)/2, {x, 0, 20}], x] (* _Benedict W. J. Irwin_, Oct 19 2016 *)
%Y A277247 Cf. A000984, A002458, A036910, A110145.
%K A277247 nonn,easy
%O A277247 0,3
%A A277247 _Vladimir Reshetnikov_, Oct 06 2016