%I #37 Apr 23 2016 14:34:12
%S 1,1,1,1,2,6,16,37,83,199,512,1343,3488,9011,23488,62094,165738,
%T 444160,1193146,3216436,8709766,23683846,64611879,176730460,484593740,
%U 1332018207,3669981318,10133197561,28032766982,77688769031,215665451243,599644845226
%N Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 3).
%C Lengths of descents are unrestricted.
%H Alois P. Heinz, <a href="/A212383/b212383.txt">Table of n, a(n) for n = 0..1000</a>
%H Vaclav Kotesovec, <a href="http://oeis.org/A212382/a212382.pdf">Asymptotic of subsequences of A212382</a>
%F G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^3).
%F Recurrence: 27*(n-2)*n*(n+1)*(7315*n^5 - 103740*n^4 + 581321*n^3 - 1621860*n^2 + 2283420*n - 1319472)*a(n) = 18*n*(43890*n^7 - 732165*n^6 + 5094566*n^5 - 19117342*n^4 + 41486098*n^3 - 51106565*n^2 + 31574358*n - 6550776)*a(n-1) - 6*(197505*n^8 - 3591000*n^7 + 27966247*n^6 - 121632688*n^5 + 321488424*n^4 - 523324472*n^3 + 503478200*n^2 - 254789904*n + 49688640)*a(n-2) + 3*(395010*n^8 - 7774515*n^7 + 66666259*n^6 - 325204119*n^5 + 983833331*n^4 - 1877089982*n^3 + 2181330624*n^2 - 1388626960*n + 361350720)*a(n-3) + 3*(n-4)*(460845*n^7 - 7918155*n^6 + 56450863*n^5 - 217083355*n^4 + 489279096*n^3 - 653902178*n^2 + 488647076*n - 160632720)*a(n-4) - 3*(n-5)*(n-4)*(43890*n^6 - 600495*n^5 + 3490586*n^4 - 10880637*n^3 + 18337012*n^2 - 14599068*n + 3573072)*a(n-5) - 23*(n-6)*(n-5)*(n-4)*(7315*n^5 - 67165*n^4 + 239511*n^3 - 427187*n^2 + 405278*n - 173016)*a(n-6). - _Vaclav Kotesovec_, Aug 18 2013
%F a(n) ~ c*d^n/n^(3/2), where d = 2.917020449755... is the root of the equation 23 + 18*d - 189*d^2 - 162*d^3 + 162*d^4 - 108*d^5 + 27*d^6 = 0 and c = 0.415028509255451481644332... - _Vaclav Kotesovec_, Aug 18 2013
%F a(n) = Sum_{k=0..n} binomial(2*k-n-1,n-k)*binomial(n+1,3*k-2*n)/(n+1). - _Vladimir Kruchinin_, Mar 05 2016
%F G.f. is series_reversion(B(x))/x where B(x) = x/(1 + x + x^4 + x^7 + x^10 + ...) = x/(1+x/(1-x^3)); for Dyck paths with allowed ascent lengths {u_1, u_2, ...} use B(x) = x/( 1 + sum(k>=1, x^{u_k} ) ). - _Joerg Arndt_, Apr 23 2016
%e a(0) = 1: the empty path.
%e a(1) = 1: UD.
%e a(2) = 1: UDUD.
%e a(3) = 1: UDUDUD.
%e a(4) = 2: UDUDUDUD, UUUUDDDD.
%e a(5) = 6: UDUDUDUDUD, UDUUUUDDDD, UUUUDDDDUD, UUUUDDDUDD, UUUUDDUDDD, UUUUDUDDDD.
%p b:= proc(x, y, u) option remember;
%p `if`(x<0 or y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p `if`(u, add(b(x-(3*t+1), y, false), t=0..(x-1)/3), 0)))
%p end:
%p a:= n-> b(n$2, true):
%p seq(a(n), n=0..40);
%p # second Maple program
%p a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^3), A), x, n+1), x, n):
%p seq(a(n), n=0..40);
%t For[A = 1; n = 1, n <= 32, n++, A = (1-(x*A)^3)/(1-x-(x*A)^3) + O[x]^n]; CoefficientList[A, x] (* _Jean-François Alcover_, Apr 23 2016 *)
%o (Maxima)
%o a(n):=sum(binomial(2*k-n-1,n-k)*binomial(n+1,3*k-2*n),k,0,n)/(n+1); /* _Vladimir Kruchinin_, Mar 05 2016 */
%o (PARI) a(n) = sum(k=0, n, binomial(2*k-n-1,n-k)*binomial(n+1,3*k-2*n)) /(n+1); \\ _Michel Marcus_, Mar 05 2016
%o (PARI) x='x+O('x^66); Vec( serreverse( x/(1+x/(1-x^3)) ) / x ) \\ _Joerg Arndt_, Apr 23 2016
%Y Column k=3 of A212382.
%K nonn
%O 0,5
%A _Alois P. Heinz_, May 12 2012