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.

User: Florian Schager

Florian Schager's wiki page.

Florian Schager has authored 4 sequences.

A369982 Number of Dyck bridges with resets from any height to zero from (0,0) to (n,0).

Original entry on oeis.org

1, 1, 5, 11, 39, 105, 335, 965, 2965, 8755, 26517, 79047, 238065, 712347, 2140473, 6414555, 19256535, 57743865, 173280215, 519743405, 1559414971, 4677875401, 14034331635, 42101584041, 126307456279, 378916960525, 1136761282175, 3410263045325, 10230829252575
Offset: 0

Views

Author

Florian Schager, Feb 07 2024

Keywords

Comments

A Dyck bridge is a lattice path with steps U = (1,1) and D = (1,-1) that is allowed to go below the x-axis and ends at altitude 0.
A reset to zero is a step R = (1,-h) at altitude h for any integer h.

Examples

			For n = 3 the a(3) = 11 solutions are UUR, UDR, URR, DUR, DDR, DRR, RUD, RUR, RDU, RDR, RRR.

Links

Crossrefs

Cf. A369316 (for a different model of resets to zero).

Programs

  • Maple
    K := 1 - z*(u + 1/u);
v1, u1 := solve(K, u);
B := -z*diff(v1, z)/v1;
W := 1/(1 - 2*z);
series(B/(-W*z + 1), z, 30);
# second Maple program:
b:= proc(x, y) option remember; `if`(x=0, `if`(y=0, 1, 0),
      b(x-1, 0)+b(x-1, abs(y-1))+b(x-1, y+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..32);  # Alois P. Heinz, Feb 07 2024
  • Mathematica
    b[x_, y_] := b[x, y] = If[x == 0, If[y == 0, 1, 0],
   b[x - 1, 0] + b[x - 1, Abs[y - 1]] + b[x - 1, y + 1]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Feb 22 2025, after Alois P. Heinz *)

Formula

G.f.: (2*z-1)/((3*z-1)*sqrt(1-4*z^2)).
a(n) ~ 3^n/sqrt(5).

A369432 Number of Dyck excursions with catastrophes from (0,0) to (n,0).

Original entry on oeis.org

1, 1, 3, 6, 16, 37, 95, 230, 582, 1434, 3606, 8952, 22446, 55917, 140007, 349374, 874150, 2183230, 5460506, 13643972, 34118328, 85270626, 213205958, 532926716, 1332420796, 3330739972, 8327221380, 20816939100, 52043684970, 130105200765, 325267849335, 813155081070
Offset: 0

Views

Author

Florian Schager, Jan 23 2024

Keywords

Comments

A Dyck excursion is a lattice path with steps U = (1,1) and D = (1,-1) that does not go below the x-axis and ends at the x-axis.
A catastrophe is a step C = (1,-k) from altitude k to altitude 0 for k >= 0.

Examples

			For n = 3 the a(3) = 6 solutions are UUC, UDC, UCC, CUD, CUC, CCC.
For n = 4 the a(4) = 16 solutions are UUUC, UUDD, UUDC, UUCC, UDUD, UDUC, UDCC, UCUD, UCUC, UCCC, CUUC, CUDC, CUCC, CCUD, CCUC, CCCC.

Links

Crossrefs

Cf. A054341 (Dyck meanders with catastrophes).
Cf. A224747 (different model of catastrophes).

Programs

  • Maple
    u1 := solve(1 - z*(1/u + u), u)[2];
M := (1 - u1)/(1 - 2*z);
E := u1/z;
F := E/(-M*z + 1);
series(F, z, 33);
# second Maple program:
b:= proc(x, y) option remember; `if`(x=0, `if`(y=0, 1, 0),
      b(x-1, 0)+`if`(y>0, b(x-1, y-1), 0)+b(x-1, y+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..31);  # Alois P. Heinz, Jan 23 2024
  • Mathematica
    b[x_, y_] := b[x, y] = If[x == 0, If[y == 0, 1, 0], b[x-1, 0] + If[y > 0, b[x-1, y-1], 0] + b[x-1, y+1]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Jul 24 2025, after Alois P. Heinz *)
  • PARI
    my(N=44,z='z+O('z^N)); Vec((1 - sqrt(1 -4*z^2))*(2*z - 1)/(z^2*(6*z - 3 + sqrt(1 - 4*z^2))))

Formula

G.f.: (1 - sqrt(1 - 4*z^2))*(2*z - 1)/(z^2*(6*z - 3 + sqrt(1 - 4*z^2))).
a(n) ~ 3/8*(5/2)^n.

A369436 Number of 2-Motzkin meanders with catastrophes of length n.

Original entry on oeis.org

1, 3, 10, 36, 136, 529, 2095, 8393, 33885, 137547, 560544, 2291181, 9386584, 38525224, 158350133, 651645511, 2684323326, 11066714500, 45656997415, 188475894929, 778444106703, 3216562337420, 13296099775859, 54979806370840, 227410731720624, 940875886301665
Offset: 0

Views

Author

Florian Schager, Jan 23 2024

Keywords

Comments

A 2-Motzkin meander is a lattice path that does not go below the x-axis. with steps U = (1,1), D = (1,-1) and two copies R = (1,0) and B = (1,0), i.e. red and blue level steps.
A catastrophe is a step C = (1,-k) from altitude k to altitude 0 for k > 1.

Examples

			For n = 2 the a(2) = 10 solutions are UU, UR, UB, UD, RU, RR, RB, BU, BR, BB.
For n = 3 the a(3) = 36 solutions are UUU, UUR, UUB, UUD, UUC, URU, URR, URB, URD, UBU, UBR, UBB, UBD, UDU, UDR, UDB, RUU, RUR, RUB, RUD, RRU, RRR, RRB, RBU, RBR, RBB, BUU, BUR, BUB, BUD, BRU, BRR, BRB, BBU, BBR, BBB.

Links

Crossrefs

Cf. A274115 (Dyck meanders).
Cf. A054391 (Motzkin meanders).

Programs

  • Maple
    K := 1 - z*(1/u + 2 + u);
u1 := solve(K, u)[2];
M := (1 - u1)/(1 - 4*z);
E := u1/z;
M1 := z*E^2;Q := z*(M - E - M1);
series(M/(1 - Q), z, 30);
  • PARI
    my(N=44,z='z+O('z^N),S=sqrt(1-4*z)); Vec((1 - 4*z - S)*z/(5*S*z^2 + 12*z^3 - 5*z*S - 15*z^2 + S + 7*z - 1))

Formula

G.f.: (1 - 4*z - sqrt(1 - 4*z))*z/(5*sqrt(1 - 4*z)*z^2 + 12*z^3 - 5*z*sqrt(-4*z + 1) - 15*z^2 + sqrt(-4*z + 1) + 7*z - 1).
D-finite with recurrence n*a(n) +4*(-3*n+2)*a(n-1) +(53*n-70)*a(n-2) +(-107*n+210)*a(n-3) +(101*n-268)*a(n-4) +18*(-2*n+7)*a(n-5)=0. - R. J. Mathar, Jan 28 2024

A369316 Number of Dyck bridges with resets to zero from (0,0) to (n,0).

Original entry on oeis.org

1, 0, 2, 2, 8, 14, 40, 84, 216, 486, 1200, 2780, 6744, 15836, 38096, 90056, 215728, 511750, 1223136, 2907052, 6939544, 16511028, 39386384, 93768696, 223589648, 532502748, 1269433376, 3023953560, 7207744496, 17172061944, 40926792224, 97513876880, 232395416672
Offset: 0

Views

Author

Florian Schager, Jan 19 2024

Keywords

Comments

A Dyck bridge is a lattice path with steps U = (1,1) and D = (1,-1) that is allowed to go below the x-axis and ends at altitude 0.
A reset to zero is a step R = (1,-h) at altitude h for |h| > 1.

Examples

			For n = 4 the a(4) = 8 paths are UUUR, UUDD, UDUD, UDDU, DUUD, DUDU, DDUU, DDDR.

Links

Crossrefs

Cf. A224747 (Dyck excursions).

Programs

  • Maple
    K := 1 - z*(u + 1/u);
v1, u1 := solve(K, u);
B := -z*diff(v1, z)/v1;
W := 1/(1 - 2*z);
W1 := -z*diff(v1, z)/v1^2;
Wminus1 := z*diff(u1, z);
Q := z*(W - B - W1 - Wminus1);
series(B/(1 - Q), z, 40);
# second Maple program:
b:= proc(x, y) option remember; `if`(x=0, `if`(y=0, 1, 0),
      `if`(y>1, b(x-1, 0), 0)+b(x-1, abs(y-1))+b(x-1, y+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..32);  # Alois P. Heinz, Jan 19 2024
  • Mathematica
    b[x_, y_] := b[x, y] = If[x == 0, If[y == 0, 1, 0],
   If[y > 1, b[x - 1, 0], 0] + b[x - 1, Abs[y - 1]] + b[x - 1, y + 1]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Feb 23 2025, after Alois P. Heinz *)
  • PARI
    seq(n) = my(r=sqrt(1 - 4*x^2 + O(x*x^n))); Vec((1 - 2*x)*(1 + r)^2/(2*(1 - 2*x - 2*x^2 + 2*x^3)*r + 2 - 4*x - 8*x^2 + 12*x^3 + 8*x^4)) \\ Andrew Howroyd, Jan 19 2024

Formula

G.f.: -(2*z - 1)*(1 + sqrt(-4*z^2 + 1))^2/((4*z^3 - 4*z^2 - 4*z + 2)*sqrt(-4*z^2 + 1) + 8*z^4 + 12*z^3 - 8*z^2 - 4*z + 2).
a(n) = (4*(2*n-5)*a(n-2) +4*(n-1)*a(n-3) -16*(n-4)*a(n-4) -16*(n-4)*a(n-5))/(n-1) for n>=5. - Alois P. Heinz, Jan 20 2024

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.