cp's OEIS Frontend

A128751 Number of ascents of length at least 2 in all skew Dyck paths of semilength n.

%I A128751 #10 Jul 23 2017 12:17:44
%S A128751 1,1,1,2,1,9,1,29,6,1,83,53,1,226,294,22,1,602,1319,297,1,1588,5244,
%T A128751 2362,90,1,4171,19302,14464,1649,1,10935,67379,75505,17155,394,1,
%U A128751 28645,226321,353721,133395,9153,1,75012,738324,1532222,862950,117903,1806,1
%N A128751 Number of ascents of length at least 2 in all skew Dyck paths of semilength n.
%C A128751 A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of its steps. An ascent in a path is a maximal sequence of consecutive U steps.
%C A128751 Row sums yield A002212.
%H A128751 E. Deutsch, E. Munarini, S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
%F A128751 T(n,0) = 1.
%F A128751 Sum_{k>=0} k*T(n,k) = A128752(n).
%F A128751 G.f.: G = G(t,z) satisfies z(1 - z + tz)G^2 - (1 - z + z^2 - tz^2)G + 1 - z = 0.
%e A128751 T(4,2)=6 because we have (UU)DD(UU)DD, (UU)DD(UU)DL, (UU)D(UU)LLL, (UU)D(UU)DLD, (UU)D(UU)DDL and (UU)D(UU)DLL (the ascents of length at least 2 are shown between parentheses).
%e A128751 Triangle starts:
%e A128751   1;
%e A128751   1;
%e A128751   1,   2;
%e A128751   1,   9;
%e A128751   1,  29,   6;
%e A128751   1,  83,  53;
%e A128751   1, 226, 294,  22;
%p A128751 eq:=z*(1-z+t*z)*G^2-(1-z+z^2-t*z^2)*G+1-z=0: G:=RootOf(eq,G): Gser:=simplify(series(G,z=0,18)): for n from 0 to 15 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 15 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form
%Y A128751 Cf. A002212, A128752.
%K A128751 nonn,tabf
%O A128751 0,4
%A A128751 _Emeric Deutsch_, Mar 31 2007