A132887 Number of symmetric paths in the first quadrant, from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0).

1, 1, 3, 2, 8, 6, 23, 17, 68, 51, 205, 154, 627, 473, 1937, 1464, 6032, 4568, 18900, 14332, 59519, 45187, 188211, 143024, 597241, 454217, 1900821, 1446604, 6065180, 4618576, 19396027, 14777451, 62148628, 47371177, 199481503, 152110326, 641275603, 489165277, 2064376454

Offset

0,3

Comments

The number of paths in the first quadrant, from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0) is A128720(n).

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

a(2n+1) = A059398(n); a(2n) = A059398(n-1) + A059398(n).

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

D-finite with recurrence (n+2)*a(n) +n*a(n-1) +(-n-6)*a(n-2) -2*n*a(n-3) +7*(-n+2)*a(n-4) +5*(-n+4)*a(n-5) +3*(-n+6)*a(n-6) +2*(n-8)*a(n-7) +(3*n-26)*a(n-8) +(n-8)*a(n-9) +(n-10)*a(n-10)=0. - R. J. Mathar, Oct 08 2016

Example

a(4)=8 because we have hhhh, hHh, HH, hUDh, UDUD, UhhD, UHD and UUDD.

Maple

G:=(2*(1+z+z^2))/(1-3*z^2-z^4+sqrt((1+z^2-z^4)*(1-3*z^2-z^4))): Gser:=series(G, z=0, 35): seq(coeff(Gser, z, n), n=0..30);

Prog

(PARI) seq(n) = {my(z='x); Vec(2*(1+z+z^2)/(1-3*z^2-z^4+sqrt((1+z^2-z^4)*(1-3*z^2-z^4) + O(z*z^n))))} \\ Andrew Howroyd, Nov 07 2025

Crossrefs

Cf. A128720, A059398.

Sequence in context: A122297 A073283 A117822 * A364317 A092174 A245781

Adjacent sequences: A132884 A132885 A132886 * A132888 A132889 A132890

Keyword

nonn

Author

Emeric Deutsch, Sep 05 2007

Extensions

More terms from Andrew Howroyd, Nov 07 2025

Status

approved