cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A129156 Number of primitive Dyck factors in all skew Dyck paths of semilength n.

Original entry on oeis.org

0, 1, 3, 10, 36, 136, 532, 2139, 8796, 36859, 156946, 677514, 2959669, 13063493, 58184838, 261230814, 1181144792, 5374078726, 24588562675, 113067256235, 522270436044, 2422244159067, 11275548912967, 52663412854571
Offset: 0

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Author

Emeric Deutsch, Apr 02 2007

Keywords

Comments

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 primitive Dyck factor is a subpath of the form UPD that starts on the x-axis, P being a Dyck path.

Examples

			a(2)=3 because in all skew Dyck paths of semilength 3, namely (UD)(UD), (UUDD) and UUDL, we have altogether 3 primitive Dyck factors (shown between parentheses).

Links

Crossrefs

Cf. A129154, A129157, A129158.

Programs

  • Maple
    G:=(3-3*z-sqrt(1-6*z+5*z^2))*(1-sqrt(1-4*z))/(1+z+sqrt(1-6*z+5*z^2))^2: Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..27);
  • Mathematica
    CoefficientList[Series[(3-3*x-Sqrt[1-6*x+5*x^2])*(1-Sqrt[1-4*x])/ (1+x+Sqrt[1-6*x+5*x^2])^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
  • PARI
    z='z+O('z^25); concat([0], Vec((3-3*z-sqrt(1-6*z+5*z^2))*(1-sqrt(1-4*z))/(1 +z + sqrt(1 - 6*z + 5*z^2))^2)) \\ G. C. Greubel, Feb 09 2017

Formula

a(n) = Sum_{k=0,..,n} k*A129154(n,k).
a(n) = A128742(n) - A129158(n).
G.f.: (3-3*z-sqrt(1-6*z+5*z^2))*(1-sqrt(1-4*z))/(1 +z + sqrt(1 - 6*z + 5*z^2))^2.
a(n) ~ (5-sqrt(5)) * 5^(n+3/2) / (36*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

A129157 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k primitive non-Dyck factors (n>=0; 0<=k<=floor((n+1)/3)).

Original entry on oeis.org

1, 1, 2, 1, 5, 5, 14, 22, 42, 94, 1, 132, 400, 11, 429, 1709, 81, 1430, 7351, 503, 1, 4862, 31857, 2851, 17, 16796, 139100, 15297, 176, 58786, 611781, 79228, 1440, 1, 208012, 2709230, 400694, 10259, 23, 742900, 12075248, 1993226, 66774, 307, 2674440
Offset: 0

Views

Author

Emeric Deutsch, Apr 02 2007

Keywords

Comments

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 primitive non-Dyck factor is a subpath of the form UPD, P being a skew Dyck path with at least one L step, or of the form UPL, P being any nonempty skew Dyck path.

Examples

			T(3,1) = 5 because we have UD(UUDL), (UUUDLD), (UUDUDL), (UUUDDL) and (UUUDLL);
T(5,2) = 1 because we have (UUUDLD)(UUDL) (the primitive non-Dyck factors are shown between parentheses).
Triangle starts:
    1;
    1;
    2,   1;
    5,   5;
   14,  22;
   42,  94,  1;
  132, 400, 11;

Links

  • E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2192-2203

Crossrefs

Cf. A000108, A002212, A129154, A129158.

Programs

  • Maple
    G:=(2+t-3*t*z-t*sqrt(1-6*z+5*z^2))/(1+t*z+(1-t)*sqrt(1-4*z)+t*sqrt(1-6*z+5*z^2)):
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+1)/3)) od;
# yields sequence in triangular form

Formula

G.f.: G(t,z) = [1+tz(g-1)]/[1-tz(g-C)-zC], where g=1+zg^2+z(g-1) = [1-z-sqrt(1- 6z+5z^2)]/(2z) and C=1+zC^2=[1-sqrt(1-4z)]/(2z) is the Catalan function.
Row n has 1+floor((n+1)/3) terms (n>=1).
Row sums yield A002212.
T(n,0) = binomial(2*n,n)/(n+1) = A000108(n) (the Catalan numbers).
Sum_{k>=0} k*T(n,k) = A129158(n).

A129155 Number of skew Dyck paths of semilength n that have no primitive Dyck factors.

Original entry on oeis.org

1, 0, 1, 4, 15, 59, 241, 1011, 4326, 18797, 82685, 367410, 1646494, 7432270, 33761322, 154213566, 707882503, 3263713148, 15107319268, 70182332975, 327111450097, 1529226524057, 7168880978609, 33693179852563
Offset: 0

Views

Author

Emeric Deutsch, Apr 02 2007

Keywords

Comments

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 primitive Dyck factor is a subpath of the form UPD that starts on the x-axis, P being a Dyck path.

Examples

			a(3)=4 because we have UUUDLD, UUDUDL, UUUDDL and UUUDLL.

Links

Crossrefs

Cf. A129154, A129157.

Programs

  • Maple
    G:=(3-3*z-sqrt(1-6*z+5*z^2))/(2+z-sqrt(1-4*z)+sqrt(1-6*z+5*z^2)): Gser:=series(G,z=0,32): seq(coeff(Gser,z,n),n=0..28);
  • Mathematica
    CoefficientList[Series[(3-3*x-Sqrt[1-6*x+5*x^2])/(2+x-Sqrt[1-4*x]+Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
  • PARI
    z='z+O('z^50); Vec((3-3*z-sqrt(1-6*z+5*z^2))/(2+z-sqrt(1-4*z)+sqrt(1-6*z+5*z^2))) \\ G. C. Greubel, Mar 20 2017

Formula

a(n) = A129154(n,0).
G.f.: (3-3*z-sqrt(1-6*z+5*z^2))/(2+z-sqrt(1-4*z)+sqrt(1-6*z+5*z^2)).
a(n) ~ (475 + 697*sqrt(5)) * 5^n / (3364*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014
Showing 1-3 of 3 results.

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