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A212384 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 4).

%I A212384 #28 Apr 15 2017 00:43:10
%S A212384 1,1,1,1,1,2,7,22,57,128,268,573,1343,3434,9038,23374,58649,144400,
%T A212384 355992,892336,2280020,5892301,15253305,39347067,101177783,260255812,
%U A212384 671941182,1743500452,4542147622,11858732144,30983904244,80982376879,211831943129,554905957520
%N A212384 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 4).
%C A212384 Lengths of descents are unrestricted.
%C A212384 The radius of convergence of g.f. A(x) is r = 4*(1-2*s+s^2)/(s*(4*s-3)) = 0.36467312501521477251..., where s = A(r) is described below. - _Vaclav Kotesovec_, Mar 21 2014
%H A212384 Alois P. Heinz, <a href="/A212384/b212384.txt">Table of n, a(n) for n = 0..1000</a>
%H A212384 Vaclav Kotesovec, <a href="/A212384/a212384.txt">Recurrence (of order 8)</a>
%H A212384 Vaclav Kotesovec, <a href="http://oeis.org/A212382/a212382.pdf">Asymptotic of subsequences of A212382</a>
%F A212384 G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^4).
%F A212384 a(n) ~ (s*(4*s-3))^(n+3/2) / (sqrt(Pi) * sqrt(5*s-3) * n^(3/2) * 2^(2*n+9/2) * (s-1)^(2*n+7/2)), where s = 1.880470225526517115847397... is the root of the equation 283 - 2156*s + 7312*s^2 - 14400*s^3 + 17920*s^4 - 14336*s^5 + 7168*s^6 - 2048*s^7 + 256*s^8 = 0. - _Vaclav Kotesovec_, Mar 21 2014
%F A212384 a(n) = Sum_{k=0..n} (binomial(3*k-2*n-1,n-k)*binomial(n+1,4*k-3*n))/(n+1). - _Vladimir Kruchinin_, Mar 05 2016
%e A212384 a(0) = 1: the empty path.
%e A212384 a(1) = 1: UD.
%e A212384 a(5) = 2: UDUDUDUDUD, UUUUUDDDDD.
%e A212384 a(6) = 7: UDUDUDUDUDUD, UDUUUUUDDDDD, UUUUUDDDDDUD, UUUUUDDDDUDD, UUUUUDDDUDDD, UUUUUDDUDDDD, UUUUUDUDDDDD.
%p A212384 b:= proc(x, y, u) option remember;
%p A212384       `if`(x<0 or  y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p A212384       `if`(u, add (b(x-(4*t+1), y, false), t=0..(x-1)/4), 0)))
%p A212384     end:
%p A212384 a:= n-> b(n$2, true):
%p A212384 seq(a(n), n=0..40);
%p A212384 # second Maple program:
%p A212384 a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^4), A), x, n+1), x, n):
%p A212384 seq(a(n), n=0..40);
%t A212384 a[n_] := Sum[Binomial[3k-2n-1, n-k]*Binomial[n+1, 4k-3n], {k, 0, n}]/(n+1);
%t A212384 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Apr 03 2017, after _Vladimir Kruchinin_ *)
%o A212384 (Maxima)
%o A212384 a(n):=sum(binomial(3*k-2*n-1,n-k)*binomial(n+1,4*k-3*n),k,0,n)/(n+1); /* _Vladimir Kruchinin_, Mar 05 2016 */
%o A212384 (PARI) a(n) = sum(k=0, n, binomial(3*k-2*n-1,n-k)*binomial(n+1,4*k-3*n))/(n+1); \\ _Michel Marcus_, Mar 05 2016
%Y A212384 Column k=4 of A212382.
%K A212384 nonn
%O A212384 0,6
%A A212384 _Alois P. Heinz_, May 12 2012