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%I #10 Jul 23 2017 12:17:44
%S 1,1,1,2,1,9,1,29,6,1,83,53,1,226,294,22,1,602,1319,297,1,1588,5244,
%T 2362,90,1,4171,19302,14464,1649,1,10935,67379,75505,17155,394,1,
%U 28645,226321,353721,133395,9153,1,75012,738324,1532222,862950,117903,1806,1
%N Number of ascents of length at least 2 in all skew Dyck paths of semilength n.
%C 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 Row sums yield A002212.
%H 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 T(n,0) = 1.
%F Sum_{k>=0} k*T(n,k) = A128752(n).
%F G.f.: G = G(t,z) satisfies z(1 - z + tz)G^2 - (1 - z + z^2 - tz^2)G + 1 - z = 0.
%e 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 Triangle starts:
%e 1;
%e 1;
%e 1, 2;
%e 1, 9;
%e 1, 29, 6;
%e 1, 83, 53;
%e 1, 226, 294, 22;
%p 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 Cf. A002212, A128752.
%K nonn,tabf
%O 0,4
%A _Emeric Deutsch_, Mar 31 2007
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