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.

A110122 Number of Delannoy paths of length n with no EE's crossing the line y = x (i.e., no two consecutive E steps from the line y = x+1 to the line y = x-1).

Original entry on oeis.org

1, 3, 12, 53, 247, 1192, 5897, 29723, 152020, 786733, 4111295, 21661168, 114925697, 613442227, 3291704108, 17745496453, 96062011319, 521943400056, 2845404909129, 15558847792747, 85311186002036, 468951179698653, 2583765541267647, 14266052382826208
Offset: 0

Views

Author

Emeric Deutsch, Jul 13 2005

Keywords

Comments

A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).
Column 0 of A110121.

Examples

			a(2) = 12 because, among the 13 (=A001850(2)) Delannoy paths of length 2, only NEEN has an EE crossing the line y=x.

Links

Crossrefs

Cf. A001850, A110121.

Programs

  • Maple
    R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=1/((1-z*R)^2-z): Gser:=series(G,z=0,27): 1,seq(coeff(Gser,z^n),n=1..24);
  • Mathematica
    Flatten[{1, RecurrenceTable[{(2*n^2+9*n+7)*a[n]-(26*n^2+135*n+151) *a[n+1]+(88*n^2+528*n+746)*a[n+2]-(26*n^2+177*n+277)*a[n+3]+(2*n^2+15*n+25)*a[n+4]==0,a[1]==3,a[2]==12,a[3]==53,a[4]==247}, a, {n, 25}]}] (* Vaclav Kotesovec, Sep 09 2012 *)
  • Maxima
    a(n):=sum((k+1)/(n+1)*sum(binomial(n+1,i)*binomial(2*n-k-i,n),i,0,n-k) *fib(k+1),k,0,n); /* Vladimir Kruchinin, Apr 18 2011 */

Formula

G.f.: 1/((1-zR)^2-z), where R = 1 + zR + zR^2 = (1 - z - sqrt(1 - 6z + z^2))/(2z) is the g.f. of the large Schroeder numbers (A006318).
a(n) = (1/(n+1))Sum_{k=0..n} (k+1) * Sum_{i=0..n-k} binomial(n+1, i)*binomial(2*n-k-i, n) * A000045(k+1). - Vladimir Kruchinin, Apr 18 2011
Recurrence: (2*n^2+9*n+7)*a(n) - (26*n^2+135*n+151)*a(n+1) + (88*n^2+528*n+746)*a(n+2) - (26*n^2+177*n+277)*a(n+3) + (2*n^2+15*n+25)*a(n+4)=0. - Vaclav Kotesovec, Sep 08 2012
a(n) ~ (10+7*sqrt(2))*sqrt((3*sqrt(2)-4)/Pi) * (3+2*sqrt(2))^n/n^(3/2). - Vaclav Kotesovec, Dec 11 2012

Extensions

Minor edits by Vaclav Kotesovec, Mar 31 2014

A110127 Number of EE's crossing the line y = x (i.e., two consecutive E steps from the line y = x+1 to the line y = x-1) in all Delannoy paths of length n.

Original entry on oeis.org

0, 0, 1, 10, 75, 508, 3277, 20566, 126871, 773688, 4679769, 28136546, 168395235, 1004239156, 5971820709, 35429993390, 209800355631, 1240361694064, 7323260678065, 43187703202234, 254439363998587, 1497730375793004
Offset: 0

Views

Author

Emeric Deutsch, Jul 13 2005

Keywords

Comments

A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).
{A110127}[n+2] = conv({0, {A002002})[n]. - Tilman Neumann, Feb 05 2009

Examples

			a(2) = 1 because, among the 13 (=A001850(2)) Delannoy paths of length 2, only NEEN has an EE crossing the line y = x.

Links

Crossrefs

Cf. A001850, A110121.

Programs

  • Maple
    R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z^2*R^2/(1-6*z+z^2): Gser:=series(G,z=0,27): 0,seq(coeff(Gser,z^n),n=1..24);
  • Mathematica
    CoefficientList[Series[x^2*((1-x-Sqrt[1-6*x+x^2])/2/x)^2/(1-6*x+x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

Formula

a(n) = Sum_{k=0..floor(n/2)} k*A110121(n,k).
G.f.: z^2*R^2/(1-6z+z^2), where R = 1+zR+zR^2 = [1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers (A006318).
Recurrence: n*(2*n-5)*a(n) = 6*(4*n^2 - 13*n + 8)*a(n-1) - 4*(19*n^2 - 76*n + 75)*a(n-2) + 6*(4*n^2 - 19*n + 20)*a(n-3) - (n-4)*(2*n-3)*a(n-4). - Vaclav Kotesovec, Oct 24 2012
a(n) ~ 1/8*sqrt(2)*(3+2*sqrt(2))^n. - Vaclav Kotesovec, Oct 24 2012

A110123 Triangle read by rows: T(n,k) is the number of Delannoy paths of length n, having k EE's and NN's crossing the line y = x (i.e., two consecutive E steps from the line y = x+1 to the line y = x-1 or two consecutive N steps from the line y = x-1 to the line y = x+1).

Original entry on oeis.org

1, 3, 11, 2, 45, 16, 2, 197, 100, 22, 2, 903, 576, 174, 28, 2, 4279, 3206, 1202, 266, 34, 2, 20793, 17568, 7732, 2128, 376, 40, 2, 103049, 95592, 47676, 15452, 3408, 504, 46, 2, 518859, 518720, 286156, 105528, 27500, 5096, 650, 52, 2, 2646723, 2813514
Offset: 0

Views

Author

Emeric Deutsch, Jul 13 2005

Keywords

Comments

A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).
Row 0 has one term; row n has n terms (n > 0).
Row sums are the central Delannoy numbers (A001850).
Column 0 yields the little Schroeder numbers (A001003).

Examples

			T(2,1)=2 because we have NEEN and ENNE.
Triangle begins:
    1;
    3;
   11,   2;
   45,  16,   2;
  197, 100,  22,   2;

Links

Crossrefs

Cf. A001003, A001850, A006318, A100127, A110121.

Programs

  • Maple
    R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=simplify((1-z*R*t+z*R)/(1-z-z*R*t+z^2*R*t-z*R-z^2*R)): Gser:=simplify(series(G,z=0,14)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser,z^n) od: 1; for n from 1 to 10 do seq(coeff(t*P[n],t^k),k=1..n) od; # yields sequence in triangular form

Formula

Sum_{k=0..n-1} k*T(n,k) = 2*A110127(n).
G.f.: (1 - tzR + zR)/(1 - z - tzR + tz^2*R - zR - z^2*R), where R = 1 + zR + zR^2 = (1 - z - sqrt(1 - 6z + z^2))/(2z) is the g.f. of the large Schroeder numbers (A006318).
Showing 1-3 of 3 results.

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