A128752 -id:A128752 - cp's OEIS frontend

A129160 Sum of the semi-abscissae of the first returns to the axis over all skew Dyck paths of semilength n.

1, 4, 18, 82, 378, 1760, 8262, 39044, 185526, 885596, 4243590, 20400954, 98353278, 475322352, 2302064010, 11170370850, 54293503770, 264290420540, 1288257980310, 6287181414470, 30717958762350, 150234512678480, 735446569221810, 3603330368706640, 17668505697688098, 86698739895529300

Offset: 1

Emeric Deutsch, Apr 03 2007

nonn

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 the path is defined to be the number of its steps.

			a(2)=4 because UDUD, UUDD and UUDL yield 1+2+1=4.

G. C. Greubel, Table of n, a(n) for n = 1..1000 (terms 1..200 from Vincenzo Librandi)
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203

Cf. A129159, A128752.

G:=z-1+(1-3*z+2*z^2)/sqrt(1-6*z+5*z^2): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=1..27);

CoefficientList[Series[(1/x) (x - 1 + (1 - 3*x + 2*x^2)/Sqrt[1 - 6*x + 5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

x='x+O('x^25); Vec(x-1+(1-3*x+2*x^2)/sqrt(1-6*x+5*x^2)) \\ G. C. Greubel, Feb 09 2017

a(n) = Sum_{k=1,..,n} k*A129159(n,k).
a(n) = 2*A128752(n) for n>=2.
G.f.: x-1+(1-3*x+2*x^2)/sqrt(1-6*x+5*x^2).
Recurrence: n*(3*n-1)*a(n) = 18*(n-1)*n*a(n-1) - 5*(n-3)*(3*n+2)*a(n-2) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 6*5^(n-3/2)/sqrt(Pi*n) . - Vaclav Kotesovec, Oct 20 2012

Mathematica code corrected by Vincenzo Librandi, May 24 2013

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

Original entry on oeis.org

1, 1, 1, 2, 1, 9, 1, 29, 6, 1, 83, 53, 1, 226, 294, 22, 1, 602, 1319, 297, 1, 1588, 5244, 2362, 90, 1, 4171, 19302, 14464, 1649, 1, 10935, 67379, 75505, 17155, 394, 1, 28645, 226321, 353721, 133395, 9153, 1, 75012, 738324, 1532222, 862950, 117903, 1806, 1

Offset: 0

Emeric Deutsch, Mar 31 2007

Row sums yield A002212.

			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).
Triangle starts:
  1;
  1;
  1,   2;
  1,   9;
  1,  29,   6;
  1,  83,  53;
  1, 226, 294,  22;

E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

Cf. A002212, A128752.

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

T(n,0) = 1.
Sum_{k>=0} k*T(n,k) = A128752(n).
G.f.: G = G(t,z) satisfies z(1 - z + tz)G^2 - (1 - z + z^2 - tz^2)G + 1 - z = 0.

cp's OEIS Frontend

A129160 Sum of the semi-abscissae of the first returns to the axis over all skew Dyck paths of semilength n.

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