cp's OEIS Frontend

A134425 Number of paths of length n in the first quadrant, starting at the origin and consisting of 2 kinds of upsteps U=(1,1) (U1 and U2), 3 kinds of flatsteps F=(1,0) (F1, F2 and F3) and 1 kind of downsteps D=(1,-1).

%I A134425 #32 Feb 15 2025 20:40:12
%S A134425 1,5,27,151,861,4969,28911,169187,994329,5862925,34658691,205305423,
%T A134425 1218183669,7238062641,43055682327,256365292443,1527728176305,
%U A134425 9110460044821,54362600841963,324557242893191,1938584147698701
%N A134425 Number of paths of length n in the first quadrant, starting at the origin and consisting of 2 kinds of upsteps U=(1,1) (U1 and U2), 3 kinds of flatsteps F=(1,0) (F1, F2 and F3) and 1 kind of downsteps D=(1,-1).
%C A134425 See A134426 for the enumeration of these paths according to the ordinates of their endpoints.
%C A134425 a(n) is the number of Schroder paths of semilength n in which the (2,0)-steps that are on the horizontal axis come in 4 colors. Example: a(2)=27 because we have 4^2=16 paths of shape HHHH, 4 paths of shape HUD, 4 paths of shape UDH, 1 paths of each of the shapes UDUD, UUDD, and UHD. - _Emeric Deutsch_, May 02 2011
%H A134425 Vincenzo Librandi, <a href="/A134425/b134425.txt">Table of n, a(n) for n = 0..200</a>
%H A134425 Taras Goy and Mark Shattuck, <a href="https://doi.org/10.15330/cmp.15.2.420-436">Hessenberg-Toeplitz Matrix Determinants with Schröder and Fine Number Entries</a>, Carpathian Math. Publ., Vol. 15 (2023), No. 2, 420-436. See Theorems 1 and 2; <a href="https://arxiv.org/abs/2303.10223">Preprint</a>, arXiv:2303.10223 [math.CO], 2023.
%F A134425 G.f.: 2/(1-7z+sqrt(1-6z+z^2)). G.f.: g/(1-2zg), where g is the g.f. of the little Schroeder numbers 1,3,11,45,197,... (A001003).
%F A134425 Recurrence: (n+1)*a(n) = 3*(4*n+1)*a(n-1) - (37*n-20)*a(n-2) + 6*(n-2)*a(n-3). - _Vaclav Kotesovec_, Oct 20 2012
%F A134425 a(n) ~ 6^n/2. - _Vaclav Kotesovec_, Oct 20 2012
%F A134425 a(n) = Sum_{k=0..n}((k+1)*Sum_{j=0..n+1}(binomial(j,n-k-j)*3^(-n+k+2*j)*2^(n-j)*binomial(n+1,j)))/(n+1). - _Vladimir Kruchinin_, Mar 13 2016
%e A134425 a(2)=5 because we have F1, F2, F3, U1 and U2. a(3)=27 because we have 9 paths of shape FF, 2 paths of shape UD, 6 paths of shape FU, 6 paths of shape UF and 4 paths of shape UU.
%p A134425 G:=2/(1-7*z+sqrt(1-6*z+z^2)): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n= 0..20);
%t A134425 CoefficientList[Series[2/(1-7*x+Sqrt[1-6*x+x^2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%o A134425 (Maxima)
%o A134425 a(n):=sum((k+1)*sum(binomial(j,n-k-j)*3^(-n+k+2*j)*2^(n-j)*binomial(n+1,j),j,0,n+1),k,0,n)/(n+1); /* _Vladimir Kruchinin_, Mar 13 2016 */
%Y A134425 Cf. A001003, A134426.
%K A134425 nonn
%O A134425 0,2
%A A134425 _Emeric Deutsch_, Nov 05 2007