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%I #70 Jul 17 2023 05:38:36
%S 0,1,0,1,1,5,13,41,121,366,1100,3319,10015,30253,91433,276475,836291,
%T 2530321,7657317,23175867,70150875,212349687,642803631,1945819299,
%U 5890003539,17828324220,53961228258,163314594513,494238394601,1495593167851,4525366817455
%N a(n) = A025179(n-2) + A102839(n-4), for n >= 4, with a(0) = a(2) = 0 and a(1) = a(3) = 1.
%C a(n) + 2*A025565(n) is the number of Dyck paths of semi-length n+2 with L(D) = 4 where L(D) is the product of binomial coefficients (u_i(D)+d_i(D) choose u_i(D)), where u_i(D) is the number of up-steps between the i-th and (i+1)-st down step and d_i(D) is the number of down-steps between the i-th and (i+1)-st up step.
%H Kassie Archer and Christina Graves, <a href="https://arxiv.org/abs/2104.12664">Pattern-restricted permutations composed of 3-cycles</a>, arXiv:2104.12664 [math.CO], 2021.
%H Kassie Archer and Christina Graves, <a href="https://arxiv.org/abs/2205.09686">A new statistic on Dyck paths for counting 3-dimensional Catalan words</a>, arXiv:2205.09686 [math.CO], 2022.
%F a(n) = m(n-3) + binomial(n-3,2)*m(n-5) + 2*Sum_{i=0..n-5} (i+1)*m(i)*m(n-5-i) for n>=3, where m(n) = A001006(n) is the n-th Motzkin number.
%F a(n) ~ sqrt(n) * 3^(n - 7/2) / (4*sqrt(Pi)). - _Vaclav Kotesovec_, Jun 03 2022
%F D-finite with recurrence (n-1)*(n-69)*a(n) +(n^2+221*n-513)*a(n-1) +(-37*n^2+356*n-312)*a(n-2) +(47*n^2-859*n+2820)*a(n-3) +12*(7*n-30)*(n-6)*a(n-4)=0. - _R. J. Mathar_, Jul 17 2023
%p a:= proc(n) option remember; `if`(n<4, irem(n, 2),
%p ((3*(n-4))*(n^4+6*n^3-41*n^2+18*n+76)*a(n-2)+
%p (2*n^5+3*n^4-136*n^3+525*n^2-658*n+132)*a(n-1))/
%p ((n^4+2*n^3-53*n^2+114*n+12)*(n-1)))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, May 19 2022
%t m[n_] := m[n] = If[n == 0, 1, m[n-1] + Sum[m[k]*m[n-2-k], {k, 0, n-2}]];
%t a[n_] := Switch[n, 0|2, 0, 1|3|4, 1, _, m[n-3] + Binomial[n-3, 2]*m[n-5] + 2*Sum[(i+1)*m[i]*m[n-5-i], {i, 0, n-5}]];
%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Dec 26 2022 *)
%o (PARI) m(n) = polcoeff( ( 1 - x - sqrt((1 - x)^2 - 4 * x^2 + x^3 * O(x^n))) / (2 * x^2), n); \\ A001006
%o a(n) = if (n<=3, n%2, m(n-3) + binomial(n-3,2)*m(n-5) + 2*sum(i=0, n-5, (i+1)*m(i)*m(n-5-i))); \\ _Michel Marcus_, May 19 2022
%Y Cf. A001006, A025179, A025565, A102839.
%K nonn
%O 0,6
%A _Kassie Archer_, Apr 26 2022
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