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

1, 0, 1, 1, 3, 2, 7, 6, 19, 16, 51, 45, 141, 126, 393, 357, 1107, 1016, 3139, 2907, 8953, 8350, 25653, 24068, 73789, 69576, 212941, 201643, 616227, 585690, 1787607, 1704510, 5196627, 4969152, 15134931, 14508939, 44152809, 42422022, 128996853

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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).

Sequence is obtained by alternating A002426 and A005717.

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Phil Hanlon, Counting interval graphs, Trans. Amer. Math. Soc. 272 (1982), no. 2, 383-426.

Crossrefs

Cf. A002426, A005717.

Programs

Maple

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);

Mathematica

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 *)

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

Formula

G.f.: ((1+2*z-z^2)/sqrt(1-2*z^2-3*z^4)-1)/(2*z).

a(2*n) = A002426(n), a(2*n+1) = [A002426(n+1) - A002426(n)]/2, (A002426(n) are the central trinomial coefficients).

Extensions

Edited by Emeric Deutsch, Nov 21 2003