A033876 Expansion of 1/(2*x) * (1/(1-4*x)^(3/2)-1).

3, 15, 70, 315, 1386, 6006, 25740, 109395, 461890, 1939938, 8112468, 33801950, 140408100, 581690700, 2404321560, 9917826435, 40838108850, 167890003050, 689232644100, 2825853840810, 11572544300460, 47342226683700, 193485622098600, 790066290235950, 3223470464162676

Offset: 0

Author

N. J. A. Sloane, Jeffrey Shallit

Keywords

• nonn
• easy

Comments

a(n) is the trace of the zigzag matrix Z(n+1) (see A088961). - Paul Boddington, Nov 03 2003

The number of edges in the odd graph O_k (for k >= 2) can be computed as 0.5*(2k-1)*C(2k-2,k-1). This sequence gives the number of edges in O_k for integer values of k from k=2. - K.V.Iyer, Mar 04 2009

Apparently the number of peaks in all symmetric Dyck paths with semilength 2n+2. - David Scambler, Apr 29 2013

For n > 0, also the number of maximal and maximum cliques in the (n+2)-odd graph. - Eric W. Weisstein, Nov 30 2017

Examples

			G.f. = 3 + 15*x + 70*x^2 + 315*x^3 + 1386*x^4 + 6006*x^5 + 25740*x^6 + ...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..250
• Eric Weisstein's World of Mathematics, Maximal Clique.
• Eric Weisstein's World of Mathematics, Maximum Clique.
• Eric Weisstein's World of Mathematics, Odd Graph.

Crossrefs

Cf. A000984, A001622, A001700, A001803, A002457, A037965, A088961.

Programs

Haskell

a033876 n = sum $ zipWith (!!) zss [0..n] where
   zss = take (n+1) $ g (take (n+1) (1 : [0,0..])) where
       g us = (take (n+1) $ g' us) : g (0 : init us)
       g' vs = last $ take (2 * n + 3) $
                      map snd $ iterate h (0, vs ++ reverse vs)
   h (p,ws) = (1 - p, drop p $ zipWith (+) ([0] ++ ws) (ws ++ [0]))
-- Reinhard Zumkeller, Oct 25 2013

Magma

[(2*n+3)*Binomial(2*n+1, n) : n in [0..40]]; // Wesley Ivan Hurt, Nov 30 2017

Maple

[seq((n+2)*binomial(2*(n+2),n+2)/4, n=0..22)]; # Zerinvary Lajos, Jan 04 2007

Mathematica

Table[nn = 2 n + 1; (2 n + 1)! Coefficient[Series[Exp[x] (x^n/n!)^2/2, {x, 0, nn}], x^(2 n + 1)], {n, 30}] (* Geoffrey Critzer, Apr 19 2017 *)
Table[n Binomial[2 n, n]/4, {n, 2, 20}] (* Eric W. Weisstein, Nov 30 2017 *)
Table[(4^n Gamma[n + 3/2])/(Sqrt[Pi] Gamma[n + 1]), {n, 20}] (* Eric W. Weisstein, Nov 30 2017 *)
CoefficientList[Series[((1 - 4 x)^(-3/2) - 1)/(2 x), {x, 0, 20}], x] (* Eric W. Weisstein, Nov 30 2017 *)

PARI

x='x+O('x^66); Vec( 1/(2*x) * (1/(1-4*x)^(3/2)-1) ) \\ Joerg Arndt, May 01 2013

Formula

a(n) = (2*n+3)*binomial(2*n+1, n). - Paul Boddington, Nov 03 2003

Equals n*A000984/4, n >= 2. - Zerinvary Lajos, Jan 04 2007

For n >= 1, 1/a(n-1) = Sum_{k>=0} binomial(2*k,k)/(4^(n+k)*(n+k+1)) = int(4*t^n/sqrt(1-4*t), t=0..1/4). - Groux Roland, Jan 17 2011

G.f.: - 1/(2*x) + G(0)/(4*x), where G(k)= 1 + 1/(1 - 2*x*(2*k+3)/(2*x*(2*k+3) + (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 18 2013

a(n) = 2^(2*n+1)*binomial(n+3/2, 1/2). - Peter Luschny, May 06 2014

0 = a(n)*(16*a(n+1) - 2*a(n+2)) + a(n+1)*(-6*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Sep 17 2014

a(n-2) = n*binomial(2*n, n)/4 for n > 1. - Eric W. Weisstein, Nov 30 2017

G.f.: ((1 - 4*x)^(-3/2) - 1)/2 (by definition). - Eric W. Weisstein, Nov 30 2017

D-finite with recurrence: (n+1)*a(n) +2*(-2*n-3)*a(n-1)=0. - R. J. Mathar, Jan 28 2020

G.f.: (1F0(3/2;;4*x)-1)/(2*x). - R. J. Mathar, Jan 28 2020

From Amiram Eldar, Mar 04 2023: (Start)

Sum_{n>=0} 1/a(n) = 4*Pi/(3*sqrt(3)) - 2.

Sum_{n>=0} (-1)^n/a(n) = 2 - 8*log(phi)/sqrt(5), where phi is the golden ratio (A001622). (End)

From Mélika Tebni, Sep 04 2024: (Start)

a(n) = A037965(n+2) - A001700(n).

E.g.f.: exp(2*x)*((3+8*x)*BesselI(0, 2*x) + (1+8*x)*BesselI(1, 2*x)). (End)

a(n) = 2^n*JacobiP(n+1, 1/2, -n-1, 3). - Peter Luschny, Jan 22 2025