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 A212696 #44 Jul 16 2024 16:14:54
%S A212696 1,0,3,4,25,66,287,960,3789,13810,53240,200652,771641,2952054,
%T A212696 11386065,43910288,170007429,658979586,2560258550,9960335060,
%U A212696 38811668868,151418146704,591464244882,2312774560296,9052560751725,35464735083726,139054217427702,545635715465596
%N A212696 Central coefficient of the triangle A097609.
%H A212696 D. Kruchinin and V. Kruchinin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kruchinin/kruchinin5.html">A Method for Obtaining Generating Function for Central Coefficients of Triangles</a>, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.
%F A212696 G.f.: (12-4/sqrt(1-4*x))/(8*sqrt(12*x+2*sqrt(1-4*x)+2))+1/(2*sqrt(1-4*x)).
%F A212696 a(n) = ((n+1)*Sum_{j=0..n} C(n+2*j, n+j)*(-1)^(n-j)*C(2*n+1, n+j+1)) / (2*n+1).
%F A212696 a(n) = (n+1)*A055113(n).
%F A212696 Conjecture: 2*n*(n-1)*(2*n+1)*(5*n-8)*a(n) -(n-1)*(115*n^3-344*n^2+299*n-82) *a(n-1) -4*(2*n-3)*(5*n^3+27*n^2-74*n+30)*a(n-2) +36*(n-1)*(5*n-3)*(2*n-3)*(2*n-5) *a(n-3)=0. - _R. J. Mathar_, Oct 08 2016
%F A212696 a(n) = (-1)^n*binomial(2*n, n)*hypergeom([(n+1)/2, 1+n/2, -n], [1+n, 2+n], 4). - _Peter Luschny_, Dec 26 2017
%F A212696 From _Emanuele Munarini_, Jul 14 2024: (Start)
%F A212696 a(n) = Sum_{k=0..floor(n/2)} binomial(2*n,k)*binomial(n-k-1,k-1)*(n+1)/(2*n-k+1).
%F A212696 a(n) = Sum_{k=0..n} (-1)^k*binomial(2*n,k)*binomial(3n-2k,2*n-k)*(n+1)/(2*n-k+1).
%F A212696 a(n) = (n+1)/(2n+1)*Sum_{k=0..n} binomial(2*n+i,2*n)*trinomial(2*n+1,n-k)*(-1)^{n-k}, where trinomial(n,k) are the trinomial coefficients (A027907).
%F A212696 a(n) = Sum_{k=0..n} (-1)^k*binomial(3*n-k,n-k)*trinomial(2*n,k)*(n+k+1)/(2*n+1). (End)
%t A212696 Table[((n + 1) Sum[Binomial[n + 2 j, n + j] (-1)^(n - j) Binomial[2 n + 1, n + j + 1], {j, 0, n}])/(2 n + 1), {n, 0, 27}] (* or *)
%t A212696 CoefficientList[Series[(12 - 4/#)/(8 Sqrt[12 x + 2 # + 2]) + 1/(2 #) &@ Sqrt[1 - 4 x], {x, 0, 27}], x] (* _Michael De Vlieger_, Oct 08 2016 *)
%t A212696 a[n_] := (-1)^n Binomial[2n, n] HypergeometricPFQ[{(n+1)/2, 1+n/2, -n}, {1+n, 2+n}, 4]; Table[a[n], {n, 0, 27}] (* _Peter Luschny_, Dec 26 2017 *)
%o A212696 (PARI)
%o A212696 x='x+O('x^66);
%o A212696 gf=(12-4/sqrt(1-4*x))/(8*sqrt(12*x+2*sqrt(1-4*x)+2))+1/(2*sqrt(1-4*x));
%o A212696 Vec(Ser(gf))
%o A212696 /* _Joerg Arndt_, Jun 09 2012 */
%Y A212696 Cf. A007318, A027907, A055113, A097609.
%K A212696 nonn
%O A212696 0,3
%A A212696 _Vladimir Kruchinin_, May 24 2012