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

%I A033876 #67 Feb 16 2025 08:32:36
%S A033876 3,15,70,315,1386,6006,25740,109395,461890,1939938,8112468,33801950,
%T A033876 140408100,581690700,2404321560,9917826435,40838108850,167890003050,
%U A033876 689232644100,2825853840810,11572544300460,47342226683700,193485622098600,790066290235950,3223470464162676
%N A033876 Expansion of 1/(2*x) * (1/(1-4*x)^(3/2)-1).
%C A033876 a(n) is the trace of the zigzag matrix Z(n+1) (see A088961). - _Paul Boddington_, Nov 03 2003
%C A033876 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
%C A033876 Apparently the number of peaks in all symmetric Dyck paths with semilength 2n+2. - _David Scambler_, Apr 29 2013
%C A033876 For n > 0, also the number of maximal and maximum cliques in the (n+2)-odd graph. - _Eric W. Weisstein_, Nov 30 2017
%H A033876 Reinhard Zumkeller, <a href="/A033876/b033876.txt">Table of n, a(n) for n = 0..250</a>
%H A033876 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximalClique.html">Maximal Clique</a>.
%H A033876 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximumClique.html">Maximum Clique</a>.
%H A033876 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddGraph.html">Odd Graph</a>.
%F A033876 a(n) = (2*n+3)*binomial(2*n+1, n). - _Paul Boddington_, Nov 03 2003
%F A033876 Equals n*A000984/4, n >= 2. - _Zerinvary Lajos_, Jan 04 2007
%F A033876 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
%F A033876 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
%F A033876 a(n) = 2^(2*n+1)*binomial(n+3/2, 1/2). - _Peter Luschny_, May 06 2014
%F A033876 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
%F A033876 a(n-2) = n*binomial(2*n, n)/4 for n > 1. - _Eric W. Weisstein_, Nov 30 2017
%F A033876 G.f.: ((1 - 4*x)^(-3/2) - 1)/2 (by definition). - _Eric W. Weisstein_, Nov 30 2017
%F A033876 D-finite with recurrence: (n+1)*a(n) +2*(-2*n-3)*a(n-1)=0. - _R. J. Mathar_, Jan 28 2020
%F A033876 G.f.: (1F0(3/2;;4*x)-1)/(2*x). - _R. J. Mathar_, Jan 28 2020
%F A033876 From _Amiram Eldar_, Mar 04 2023: (Start)
%F A033876 Sum_{n>=0} 1/a(n) = 4*Pi/(3*sqrt(3)) - 2.
%F A033876 Sum_{n>=0} (-1)^n/a(n) = 2 - 8*log(phi)/sqrt(5), where phi is the golden ratio (A001622). (End)
%F A033876 From _Mélika Tebni_, Sep 04 2024: (Start)
%F A033876 a(n) = A037965(n+2) - A001700(n).
%F A033876 E.g.f.: exp(2*x)*((3+8*x)*BesselI(0, 2*x) + (1+8*x)*BesselI(1, 2*x)). (End)
%F A033876 a(n) = 2^n*JacobiP(n+1, 1/2, -n-1, 3). - _Peter Luschny_, Jan 22 2025
%e A033876 G.f. = 3 + 15*x + 70*x^2 + 315*x^3 + 1386*x^4 + 6006*x^5 + 25740*x^6 + ...
%p A033876 [seq((n+2)*binomial(2*(n+2),n+2)/4, n=0..22)]; # _Zerinvary Lajos_, Jan 04 2007
%t A033876 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 *)
%t A033876 Table[n Binomial[2 n, n]/4, {n, 2, 20}] (* _Eric W. Weisstein_, Nov 30 2017 *)
%t A033876 Table[(4^n Gamma[n + 3/2])/(Sqrt[Pi] Gamma[n + 1]), {n, 20}] (* _Eric W. Weisstein_, Nov 30 2017 *)
%t A033876 CoefficientList[Series[((1 - 4 x)^(-3/2) - 1)/(2 x), {x, 0, 20}], x] (* _Eric W. Weisstein_, Nov 30 2017 *)
%o A033876 (PARI) x='x+O('x^66); Vec( 1/(2*x) * (1/(1-4*x)^(3/2)-1) ) \\ _Joerg Arndt_, May 01 2013
%o A033876 (Haskell)
%o A033876 a033876 n = sum $ zipWith (!!) zss [0..n] where
%o A033876    zss = take (n+1) $ g (take (n+1) (1 : [0,0..])) where
%o A033876        g us = (take (n+1) $ g' us) : g (0 : init us)
%o A033876        g' vs = last $ take (2 * n + 3) $
%o A033876                       map snd $ iterate h (0, vs ++ reverse vs)
%o A033876    h (p,ws) = (1 - p, drop p $ zipWith (+) ([0] ++ ws) (ws ++ [0]))
%o A033876 -- _Reinhard Zumkeller_, Oct 25 2013
%o A033876 (Magma) [(2*n+3)*Binomial(2*n+1, n) : n in [0..40]]; // _Wesley Ivan Hurt_, Nov 30 2017
%Y A033876 Cf. A000984, A001622, A001700, A001803, A002457, A037965, A088961.
%K A033876 nonn,easy
%O A033876 0,1
%A A033876 _N. J. A. Sloane_, _Jeffrey Shallit_