cp's OEIS Frontend

A005213 Number of symmetric, reduced unit interval schemes with n+1 intervals (n>=1).

%I A005213 M2254 #18 Jan 10 2018 03:16:30
%S A005213 1,0,1,1,3,2,7,6,19,16,51,45,141,126,393,357,1107,1016,3139,2907,8953,
%T A005213 8350,25653,24068,73789,69576,212941,201643,616227,585690,1787607,
%U A005213 1704510,5196627,4969152,15134931,14508939,44152809,42422022,128996853
%N A005213 Number of symmetric, reduced unit interval schemes with n+1 intervals (n>=1).
%C A005213 Also, number of symmetric Dyck paths of semilength n with no peaks at odd level. E.g., a(4)=3 because we have UUUUDDDD, UUDDUUDD and UUDUDUDD, where U=(1,1) and D=(1,-1).
%C A005213 Sequence is obtained by alternating A002426 and A005717.
%D A005213 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005213 G. C. Greubel, <a href="/A005213/b005213.txt">Table of n, a(n) for n = 0..1000</a>
%H A005213 Phil Hanlon, <a href="http://dx.doi.org/10.1090/S0002-9947-1982-0662044-8">Counting interval graphs</a>, Trans. Amer. Math. Soc. 272 (1982), no. 2, 383-426.
%F A005213 G.f.: ((1+2*z-z^2)/sqrt(1-2*z^2-3*z^4)-1)/(2*z).
%F A005213 a(2*n) = A002426(n), a(2*n+1) = [A002426(n+1) - A002426(n)]/2, (A002426(n) are the central trinomial coefficients).
%p A005213 G:=((1+2*z-z^2)/sqrt(1-2*z^2-3*z^4)-1)/(2*z): Gser:=series(G,z=0,40): 1,seq(coeff(Gser,z^n),n=1..38);
%t A005213 CoefficientList[Series[((1 + 2*z - z^2)/Sqrt[1 - 2*z^2 - 3*z^4] - 1)/(2*z), {z, 0, 50}], z] (* _G. C. Greubel_, Mar 02 2017 *)
%o A005213 (PARI) x='x +O('x^50); Vec(((1+2*x-x^2)/sqrt(1-2*x^2-3*x^4)-1)/(2*x)) \\ _G. C. Greubel_, Mar 02 2017
%Y A005213 Cf. A002426, A005717.
%K A005213 nonn
%O A005213 0,5
%A A005213 _N. J. A. Sloane_
%E A005213 Edited by _Emeric Deutsch_, Nov 21 2003