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A033876 Expansion of 1/(2*x) * (1/(1-4*x)^(3/2)-1). 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..250

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(binomial(2*k,k)/(4^(n+k)*(n+k+1)),k=0..infinity) = 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

EXAMPLE

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

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, 1, 30}] (* Geoffrey Critzer, Apr 19 2017 *)

PROG

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

(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

CROSSREFS

Cf. A000984, A001803, A002457, A088961.

Sequence in context: A277370 A213140 A245751 * A009174 A178345 A183547

Adjacent sequences:  A033873 A033874 A033875 * A033877 A033878 A033879

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jeffrey Shallit

STATUS

approved

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Last modified June 28 16:43 EDT 2017. Contains 288839 sequences.