cp's OEIS Frontend

A274112 Number of equivalence classes of ballot paths of length n for the string ddu.

%I A274112 #33 Aug 02 2018 10:34:56
%S A274112 1,1,1,1,2,3,4,5,8,12,17,23,35,52,75,105,157,232,337,480,712,1049,
%T A274112 1529,2199,3248,4777,6976,10092,14869,21845,31937,46377,68222,100159,
%U A274112 146536,213328,313487,460023,673351,981976,1441999,2115350,3097326,4522529,6637879,9735205,14257734,20836827,30572032,44829766,65666593
%N A274112 Number of equivalence classes of ballot paths of length n for the string ddu.
%H A274112 Gheorghe Coserea, <a href="/A274112/b274112.txt">Table of n, a(n) for n = 0..300</a>
%H A274112 K. Manes, A. Sapounakis, I. Tasoulas, P. Tsikouras, <a href="http://arxiv.org/abs/1510.01952">Equivalence classes of ballot paths modulo strings of length 2 and 3</a>, arXiv:1510.01952 [math.CO], 2015, Section 3.4.
%F A274112 G.f. y satisfies: 0 = x*(x^3+x-1)*y^2 + (2*x-1)*y + 1. - _Gheorghe Coserea_, Jan 05 2017
%F A274112 G.f.: 1/(1 - x - x^4/(1 - x^4/(1 - x^4/(1 - x^4/(1 - ...))))), a continued fraction. - _Ilya Gutkovskiy_, Jul 26 2017
%F A274112 a(n) ~ 3 * (1+r^2)^(n+1) / (7 + 4*r + 8*r^2), where r = A263719 = ((9+sqrt(93))/2)^(1/3)/3^(2/3) - (2/(3*(9+sqrt(93))))^(1/3) = 0.682327803828019327369483739711048256891188581898... is the real root of the equation r^3 + r = 1. - _Vaclav Kotesovec_, Nov 27 2017
%p A274112 A274112 := proc(n)
%p A274112     add( (n-4*i+1)/(n-3*i+1)*binomial(n-2*i,i),i=0..n/4) ;
%p A274112 end proc:
%p A274112 seq(A274112(n),n=0..50) ; # _R. J. Mathar_, Jun 20 2016
%t A274112 a[n_] := Sum[(n - 4*i + 1)/(n - 3*i + 1)*Binomial[n - 2*i, i], {i, 0, n/4} ];
%t A274112 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Nov 27 2017, after _R. J. Mathar_ *)
%o A274112 (PARI)
%o A274112 x='x; y='y;
%o A274112 Fxy = x*(x^3+x-1)*y^2 + (2*x-1)*y + 1;
%o A274112 seq(N) = {
%o A274112   my(y0 = 1 + O('x^N), y1=0);
%o A274112   for (k = 1, N,
%o A274112     y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy, y), y, y0);
%o A274112     if (y1 == y0, break()); y0 = y1);
%o A274112   Vec(y0);
%o A274112 };
%o A274112 seq(51)  \\ _Gheorghe Coserea_, Jan 05 2017
%Y A274112 Cf. A274110-A274115, A263719.
%K A274112 nonn,walk
%O A274112 0,5
%A A274112 _N. J. A. Sloane_, Jun 17 2016
%E A274112 a(0)=1 prepended by _Gheorghe Coserea_, Jan 05 2017