A005213 Number of symmetric, reduced unit interval schemes with n+1 intervals (n>=1). (Formerly M2254)

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

0,5

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.

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).

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 *)

Prog

(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

Crossrefs

Cf. A002426, A005717.

Sequence in context: A364787 A082824 A088657 * A075701 A016603 A199183

Adjacent sequences: A005210 A005211 A005212 * A005214 A005215 A005216

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Edited by Emeric Deutsch, Nov 21 2003

Status

approved