A167638 - OEIS

%I #24 Feb 06 2025 22:02:10

%S 1,0,0,1,0,2,1,5,5,15,21,51,85,188,344,730,1407,2935,5831,12094,24480,

%T 50754,103995,216043,446447,930206,1934328,4043275,8448882,17716170,

%U 37166403,78163336,164520540,346935912,732317063,1548096255,3275859473

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

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

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

%F 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

%F 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

%F 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

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

%p 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);

%o (Maxima)

%o 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);

%o /* _Vladimir Kruchinin_, May 06 2018 */

%Y Cf. A167635, A167637.

%K nonn,changed

%O 0,6

%A _Emeric Deutsch_, Nov 08 2009