OFFSET
0,3
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.
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)
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
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, May 24 2012
STATUS
approved