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

Author

Emeric Deutsch, Nov 08 2009

Keywords

• nonn

Comments

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

Examples

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

Crossrefs

Cf. A167635, A167637.

Programs

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);

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 */

Formula

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

Conjecture: g.f. A(x) = 1 + (x^3)*exp(Sum_{n >= 1} g(n, x)*x^(2*n)/n), where g(n, x) = Sum_{k = 0..n} binomial(n, k)^2*(1 + x)^k. Cf. A129509. - Peter Bala, Sep 10 2024