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A033876
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Expansion of 1/(2*x) * (1/(1-4*x)^(3/2)-1).
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12
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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
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OFFSET
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0,1
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COMMENTS
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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
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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G.f. = 3 + 15*x + 70*x^2 + 315*x^3 + 1386*x^4 + 6006*x^5 + 25740*x^6 + ...
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MAPLE
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[seq((n+2)*binomial(2*(n+2), n+2)/4, n=0..22)]; # Zerinvary Lajos, Jan 04 2007
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MATHEMATICA
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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 *)
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PROG
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(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
(MAGMA) [(2*n+3)*Binomial(2*n+1, n) : n in [0..40]]; // Wesley Ivan Hurt, Nov 30 2017
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CROSSREFS
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Cf. A000984, A001803, A002457, A088961.
Sequence in context: A277370 A213140 A245751 * A291031 A009174 A178345
Adjacent sequences: A033873 A033874 A033875 * A033877 A033878 A033879
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Jeffrey Shallit
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STATUS
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approved
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