A128723 -id:A128723 - cp's OEIS frontend

A128722 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k hills (i.e., peaks at level 1) (0 <= k <= n).

1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 22, 9, 4, 0, 1, 84, 35, 12, 5, 0, 1, 334, 138, 49, 15, 6, 0, 1, 1368, 563, 198, 64, 18, 7, 0, 1, 5734, 2352, 825, 264, 80, 21, 8, 0, 1, 24480, 10015, 3504, 1121, 336, 97, 24, 9, 0, 1, 106086, 43308, 15123, 4833, 1452, 414, 115, 27, 10, 0, 1

Offset: 0

Emeric Deutsch, Mar 30 2007

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.
T(n,0) = A128723(n).
Row sums yield A002212.
Sum_{k=0..n} k*T(n,k) = A033321(n).

			T(3,1)=3 because we have (UD)UUDD, (UD)UUDL and UUDD(UD) (the hills are shown between parentheses).
Triangle starts:
   1;
   0,  1;
   2,  0,  1;
   6,  3,  0,  1;
  22,  9,  4,  0,  1;
  84, 35, 12,  5,  0,  1;

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

Cf. A002212, A033321, A128723.

g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=(1-z+z*g)/(1+z-z*g-t*z): Gser:=simplify(series(G,z=0,14)): for n from 0 to 12 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

G.f.: (1-z+zg)/(1+z-zg-tz), where g = 1+zg^2+z(g-1) = (1-z-sqrt(1-6z+5z^2))/(2z).

A354539 Number of decorated Dyck paths of length n without peaks at level 1 ending at arbitrary levels.

Original entry on oeis.org

1, 1, 1, 2, 5, 8, 18, 31, 71, 126, 290, 527, 1218, 2253, 5223, 9796, 22763, 43170, 100502, 192347, 448476, 864887, 2019121, 3919162, 9159252, 17877619, 41819003, 82021628, 192015633

Offset: 0

R. J. Mathar, Aug 17 2022

nonn

H. Prodinger, Skew Dyck paths having no peaks at level 1, JIS 25 (2022) # 22.1.16, section 2.3.

Cf. A128723 (ending at level 0).

g := (-2*z^5-3*z^4+z^3-5*z^2-3*z+4-(z^2+3*z+4)*sqrt(1-6*z^2+5*z^4))/2/z/(3+z^2)/(z^2+2*z-1) ;
taylor(%,z=0,30) ;
gfun[seriestolist](%) ;

G.f.: (-2*z^5-3*z^4+z^3-5*z^2-3*z+4-(z^2+3*z+4)*sqrt(1-6*z^2+5*z^4))/2/z/(3+z^2)/(z^2+2*z-1) .

D-finite with recurrence 12*(n+1)*a(n) +3*(-5*n-11)*a(n-1) +5*(-19*n+29)*a(n-2) +14*(5*n-4)*a(n-3) +2*(93*n-356)*a(n-4) +2*(20*n-81)*a(n-5) +2*(-22*n+217)*a(n-6) +2*(-35*n+268)*a(n-7) +2*(-27*n+182)*a(n-8) +5*(-5*n+39)*a(n-9) +5*(-n+9)*a(n-10)=0.

cp's OEIS Frontend

A128722 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k hills (i.e., peaks at level 1) (0 <= k <= n).

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