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

0,3

Links

Table of n, a(n) for n=0..27.

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.

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

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 *)

Prog

(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 */

Crossrefs

Cf. A055113, A097609.

Sequence in context: A304210 A245244 A009391 * A192346 A335739 A055348

Adjacent sequences: A212693 A212694 A212695 * A212697 A212698 A212699

Keyword

nonn

Author

Vladimir Kruchinin, May 24 2012

Status

approved