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

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(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

A129158 Number of primitive non-Dyck factors in all skew Dyck paths of semilength n.

Original entry on oeis.org

0, 0, 1, 5, 22, 96, 422, 1871, 8360, 37610, 170222, 774561, 3541487, 16263250, 74981226, 346957923, 1610847944, 7501970397, 35038158569, 164083453482, 770312822822, 3624741537711, 17093452878067, 80773023036909
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

			a(2)=1 because in all skew Dyck paths of semilength 3, namely UDUD, UUDD and (UUDL), we have altogether 1 primitive non-Dyck factor (shown between parentheses).

Links

Crossrefs

Cf. A129157, A129156, A128742.

Programs

  • Maple
    G:=(1-5*z+3*(1-z)*sqrt(1-4*z)-3*sqrt(1-6*z+5*z^2)-sqrt((1-4*z)*(1-6*z+5*z^2)))/(1+z+sqrt(1-6*z+5*z^2))^2: Gser:=series(G,z=0,32): seq(coeff(Gser,z,n),n=0..27);
  • Mathematica
    CoefficientList[Series[(1-5*x+3*(1-x)*Sqrt[1-4*x]-3*Sqrt[1-6*x+5*x^2]-Sqrt[(1-4*x)*(1-6*x+5*x^2)])/(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([0,0], Vec((1-5*z+3*(1-z)*sqrt(1-4*z)-3*sqrt(1-6*z+5*z^2) - sqrt((1-4*z)*(1-6*z+5*z^2))) /(1+z+ sqrt(1-6*z+5*z^2) )^2)) \\ G. C. Greubel, Feb 09 2017

Formula

a(n) = Sum_{k=0,..,n} k*A129157(n,k).
a(n) = A128742(n) - A129156(n).
G.f.: (1-5*z+3*(1-z)*sqrt(1-4*z)-3*sqrt(1-6*z+5*z^2) - sqrt((1-4*z)*(1-6*z+5*z^2)))/(1+z+sqrt(1-6*z+5*z^2))^2.
a(n) ~ (3*sqrt(5)+5) * 5^(1+n) / (36*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

A128741 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k returns to the x-axis (1 <= k <= n).

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 20, 11, 4, 1, 72, 42, 17, 5, 1, 274, 166, 72, 24, 6, 1, 1086, 675, 307, 111, 32, 7, 1, 4438, 2809, 1322, 506, 160, 41, 8, 1, 18570, 11913, 5752, 2296, 775, 220, 51, 9, 1, 79174, 51319, 25274, 10418, 3692, 1127, 292, 62, 10, 1, 342738, 223977, 112054
Offset: 1

Views

Author

Emeric Deutsch, Mar 31 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.
Row sums yield A002212.
T(n,1) = 2*A002212(n-1) for n >= 2 (obvious: the path of semilength n with exactly one return are of the form UPD and UPL, where P is a path of semilength n-1).
Sum_{k=1..n} k*T(n,k) = A128742(n).

Examples

			T(4,3)=4 because we have U(D)U(D)UUD(D), U(D)U(D)UUD(L), U(D)UUD(D)U(D) and UUD(D)U(D)U(D) (the return steps to the x-axis are shown between parentheses).
Triangle starts:
   1;
   2,  1;
   6,  3,  1;
  20, 11,  4,  1;
  72, 42, 17,  5,  1;

Links

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

Crossrefs

Cf. A002212, A128742.

Programs

  • Maple
    g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=(1-t*z+t*z*g)/(1-t*z*g)-1: Gser:=simplify(series(G,z=0,15)): for n from 1 to 13 do P[n]:=sort(coeff(Gser,z,n),n=1..11) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=1..n) od;

Formula

G.f.: (1 - tz + tzg)/(1 - tzg) - 1, where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z).
Column k has g.f. z^k*g^(k-1)*(2g-1).
Showing 1-3 of 3 results.

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