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.

A108447 Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have no peaks of the form ud.

Original entry on oeis.org

1, 1, 4, 20, 113, 688, 4404, 29219, 199140, 1385904, 9807820, 70364704, 510609620, 3741212535, 27639233548, 205660399220, 1539916433473, 11594310041792, 87725707127600, 666681174728724, 5086601816592432, 38948589882247968
Offset: 0

Views

Author

Emeric Deutsch, Jun 10 2005

Keywords

Comments

Column 0 of A108446.

Examples

			a(2)=4 because we have uUddd, UddUdd, UdUddd and UUdddd.

Links

Crossrefs

Cf. A000108, A027307, A108446, A108425, A108426.

Programs

  • Maple
    a:=n->(1/n)*sum(binomial(n,j)*binomial(n+2*j,j-1),j=0..n): 1, seq(a(n),n=1..25);
a := n -> `if`(n=0,1,simplify(hypergeom([1-n,(n+3)/2,(n+4)/2],[2, n+3],-4))): seq(a(n), n=0..21); # Peter Luschny, Oct 30 2015
  • Mathematica
    Flatten[{1, Table[Sum[Binomial[n, j]*Binomial[n + 2*j, j-1], {j, 0, n}]/n, {n, 1, 20}]}] (* Vaclav Kotesovec, Nov 27 2017 *)
terms = 22; g[] = 1; Do[g[x] = 1+x*g[x]*(g[x]^2+g[x]-1) + O[x]^terms // Normal, {terms}]; CoefficientList[g[x], x] (* Jean-François Alcover, Jul 19 2018 *)

Formula

a(n) = (1/n) * Sum_{j=0..n} binomial(n, j)*binomial(n+2j, j-1) (n>=1); a(0)=1.
G.f.: G satisfies G = 1 + z*G*(G^2+G-1).
a(n) = hypergeom([1-n,(n+3)/2,(n+4)/2],[2,n+3],-4) for n>=1. - Peter Luschny, Oct 30 2015
a(n) ~ sqrt((s-1) / (Pi*(1 + 3*s))) / (2*n^(3/2) * r^(n + 1/2)), where r = 0.1215851068721183026145063923222031450327682505108... and s = 1.451605962955776643742608112028547116887657025022... are real roots of the system of equations 1 + r*s*(-1 + s + s^2) = s, r*(-1 + 2*s + 3*s^2) = 1. - Vaclav Kotesovec, Nov 27 2017
O.g.f.: A(x) = (1/x) * Revert( x/c(x/(1 - x)) ), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108. - Peter Bala, Mar 08 2020
D-finite with recurrence 8*n*(2*n+1)*a(n) -6*(2*n-1)*(13*n-10)*a(n-1) +24*(4*n-7)*(2*n-5)*a(n-2) +4*(19*n-40)*(n-3)*a(n-3) -35*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 26 2022

A108426 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have k peaks of the form Ud.

Original entry on oeis.org

1, 1, 1, 3, 5, 2, 12, 28, 21, 5, 55, 165, 180, 84, 14, 273, 1001, 1430, 990, 330, 42, 1428, 6188, 10920, 10010, 5005, 1287, 132, 7752, 38760, 81396, 92820, 61880, 24024, 5005, 429, 43263, 245157, 596904, 813960, 678300, 352716, 111384, 19448, 1430, 246675
Offset: 0

Views

Author

Emeric Deutsch, Jun 03 2005

Keywords

Comments

Row sums yield A027307.
T(n,n) = A000108(n) (the Catalan numbers).
T(n,0) = A001764(n) = binomial(3n,n)/(2n+1).
Number of Ud peaks in all paths from (0,0) to (3n,0) is given by A108427.

Examples

			Example T(2,1) = 5 because we have udUdd, uUddd, Uddud, Ududd and UUdddd.
Triangle begins:
1;
1,1;
3,5,2;
12,28,21,5;
...

Links

Crossrefs

Cf. A027307, A000108, A001764, A108425, A108427.

Programs

  • Maple
    T:=(n,k)->binomial(n,k)*binomial(3*n-k,n-1)/n: print(1); for n from 1 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
  • Mathematica
    Table[If[n == 0, 1, (1/n)*Binomial[n, k]*Binomial[3 n - k, n - 1]], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Nov 29 2017 *)
  • PARI
    for(n=0,10, for(k=0,n, print1(if(n==0, 1, (1/n)*binomial(n,k) *binomial(3*n-k,n-1)), ", "))) \\ G. C. Greubel, Nov 29 2017

Formula

T(n,k) = (1/n)*binomial(n,k)*binomial(3*n-k,n-1).
G.f.: G = G(t,z) satisfies G=1+z(t+G)G^2.

A108446 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and have k peaks of the form ud.

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 20, 32, 13, 1, 113, 223, 135, 26, 1, 688, 1620, 1300, 412, 45, 1, 4404, 12064, 12050, 5350, 1030, 71, 1, 29219, 91335, 109134, 62450, 17575, 2247, 105, 1, 199140, 699689, 973077, 682234, 254625, 49210, 4438, 148, 1, 1385904, 5407744
Offset: 0

Views

Author

Emeric Deutsch, Jun 10 2005

Keywords

Comments

Row sums yield A027307. Column 0 yields A108447. T(n,n-1) = A008778(n-1) = n(n^2+6n-1)/6. Number of ud peaks in all paths from (0,0) to (3n,0) is given by A108448.

Examples

			T(2,1) = 5 because we have udUdd, uudd, Uddud, Ududd and Uuddd.
Triangle begins:
1;
1,1;
4,5,1;
20,32,13,1;
113,223,135,26,1;

Links

Crossrefs

Cf. A027307, A008778, A108447, A108448, A108425, A108426.

Programs

  • Maple
    T:=proc(n,k) if n=0 and k=0 then 1 elif n=0 then 0 elif k=n then 1 elif k=n then 1 else (1/n)*binomial(n,k)*sum(binomial(n-k,j)*binomial(n+2*j,k+j-1),j=0..n-k) fi end: for n from 0 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
  • Mathematica
    T[0, 0] = 1; T[n_, k_] := (1/n) Binomial[n, k]*Sum[Binomial[n-k, j]* Binomial[n+2j, k+j-1], {j, 0, n-k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 19 2018 *)

Formula

T(n,k) = (1/n) binomial(n, k)*sum(binomial(n-k,j)*binomial(n+2j,k+j-1), j=0..n-k).
G.f.: G = G(t,z) satisfies G = 1+z(G-1+t)G+zG^3.
Showing 1-3 of 3 results.

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