login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A212696 Central coefficient of the triangle A097609. 0
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 (list; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 19 22:05 EDT 2021. Contains 348095 sequences. (Running on oeis4.)