A167638 Number of Dyck paths of semilength n, having no ascents and no descents of length 1, and having no peaks at even level.

1, 0, 0, 1, 0, 2, 1, 5, 5, 15, 21, 51, 85, 188, 344, 730, 1407, 2935, 5831, 12094, 24480, 50754, 103995, 216043, 446447, 930206, 1934328, 4043275, 8448882, 17716170, 37166403, 78163336, 164520540, 346935912, 732317063, 1548096255, 3275859473

Offset

0,6

Comments

a(n) = A167637(n,0).

Links

Table of n, a(n) for n=0..36.

Formula

G.f.: (1 + 2z - z^3 - sqrt(1 - 4z^2 - 2z^3 + z^6))/(2z(1 + z )).

a(n) = Sum_{i=0..n} Sum_{k=1..n-i} Sum_{j=0..i-k+1} (-1)^(n-j-1)*C(j,-k-j+i+1)*C(k+j-1,k-1)*C(2*k+j-2,k+j-1)*C(n-i-1,n-k-i))/k. - Vladimir Kruchinin, May 06 2018

D-finite with recurrence (n+1)*a(n) +a(n-1) +(-4*n+7)*a(n-2) +2*(-n+2)*a(n-3) +a(n-4) -a(n-5) +(n-7)*a(n-6)=0. - R. J. Mathar, Jul 24 2022

Example

a(5)=2 because we have UUUDDUUDDD and UUUUUDDDDD.

Maple

G := ((1+2*z-z^3-sqrt(1-4*z^2-2*z^3+z^6))*1/2)/(z*(1+z)): Gser := series(G, z = 0, 40): seq(coeff(Gser, z, n), n = 0 .. 38);

Prog

(Maxima)
a(n):=sum(sum(((sum((-1)^(n-j-1)*binomial(j, -k-j+i+1)*binomial(k+j-1, k-1)*binomial(2*k+j-2, k+j-1), j, 0, -k+i+1))*binomial(n-i-1, n-k-i))/k, k, 1, n-i), i, 0, n);
/* Vladimir Kruchinin, May 06 2018 */

Crossrefs

Cf. A167635, A167637.

Sequence in context: A128731 A129157 A086905 * A317878 A209108 A269019

Adjacent sequences: A167635 A167636 A167637 * A167639 A167640 A167641

Keyword

nonn

Author

Emeric Deutsch, Nov 08 2009

Status

approved