A212696 Central coefficient of the triangle A097609.
1, 0, 3, 4, 25, 66, 287, 960, 3789, 13810, 53240, 200652, 771641, 2952054, 11386065, 43910288, 170007429, 658979586, 2560258550, 9960335060, 38811668868, 151418146704, 591464244882, 2312774560296, 9052560751725, 35464735083726, 139054217427702, 545635715465596
Offset: 0
Keywords
Links
- D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.
Programs
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Mathematica
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 *) 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 *) 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 *) -
PARI
x='x+O('x^66); gf=(12-4/sqrt(1-4*x))/(8*sqrt(12*x+2*sqrt(1-4*x)+2))+1/(2*sqrt(1-4*x)); Vec(Ser(gf)) /* Joerg Arndt, Jun 09 2012 */
Formula
G.f.: (12-4/sqrt(1-4*x))/(8*sqrt(12*x+2*sqrt(1-4*x)+2))+1/(2*sqrt(1-4*x)).
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).
a(n) = (n+1)*A055113(n).
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
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
From Emanuele Munarini, Jul 14 2024: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n,k)*binomial(n-k-1,k-1)*(n+1)/(2*n-k+1).
a(n) = Sum_{k=0..n} (-1)^k*binomial(2*n,k)*binomial(3n-2k,2*n-k)*(n+1)/(2*n-k+1).
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).
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)