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.

A034929 A triangle of Motzkin ballot numbers, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 6, 6, 4, 1, 5, 10, 13, 13, 9, 1, 6, 15, 24, 30, 30, 21, 1, 7, 21, 40, 59, 72, 72, 51, 1, 8, 28, 62, 105, 148, 178, 178, 127, 1, 9, 36, 91, 174, 276, 378, 450, 450, 323, 1, 10, 45, 128, 273, 480, 730, 980, 1158, 1158, 835, 1, 11, 55, 174, 410, 791
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Mirror image of A091836. Row sums are the Motzkin numbers (A001006). T(n,n-1)=A001006(n-2) (the Motzkin numbers). T(n,n-2)=A005554(n-1).

Examples

			Triangle begins:
[1],
[1, 1],
[1, 2, 1],
[1, 3, 3, 2],
[1, 4, 6, 6, 4],
[1, 5, 10, 13, 13, 9],
[1, 6, 15, 24, 30, 30, 21],
[1, 7, 21, 40, 59, 72, 72, 51]

References

  • M. Aigner, Motzkin numbers, Europ. J. Comb. 19 (1998), 663-675.

Crossrefs

Cf. A001006, A091836, A005554.

Formula

G.f.= 2(1+tz)/[1-2z+tz-2tz^2+sqrt(1-2tz-3t^2*z^2)].

Extensions

Edited by Emeric Deutsch, Mar 11 2004

A263917 Riordan array (f(x)^3, f(x)), where 1 + x*f^3(x)/(1 - x*f(x)) = f(x).

Original entry on oeis.org

1, 3, 1, 15, 4, 1, 85, 22, 5, 1, 519, 132, 30, 6, 1, 3330, 837, 190, 39, 7, 1, 22135, 5516, 1250, 260, 49, 8, 1, 151089, 37404, 8461, 1773, 343, 60, 9, 1, 1052805, 259280, 58550, 12324, 2422, 440, 72, 10, 1, 7458236, 1829018, 412375, 87045, 17283, 3214, 552, 85, 11, 1
Offset: 0

Views

Author

Peter Bala, Oct 29 2015

Keywords

Comments

Riordan arrays of the form (f(x)^(m+1), f(x)), where f(x) satisfies 1 + x*f^(m+1)(x)/(1 - x*f(x)) = f(x) include (modulo differences of offset) the Motzkin triangle A091836 (m = -1), the Catalan triangle A033184 (m = 0) and the Schroder triangle A091370 (m = 1). This is the case m = 2. See A263918 for the case m = 3.
The coefficients of the power series solution of the equation 1 + x*f^(m+1)(x)/(1 - x*f(x)) = f(x) appear to be given by [x^0] f(x) = 1 and [x^n] f(x) = 1/n * Sum_{k = 1..n} binomial(n,k)*binomial(n + m*k, k - 1) for n >= 1.
This triangle appears in Novelli et al., Figure 8, p. 24, where a combinatorial interpretation is given in terms of trees.

Examples

			Triangle begins:
       1
       3     1
      15     4     1
      85    22     5    1
     519   132    30    6   1
    3330   837   190   39   7  1
   22135  5516  1250  260  49  8 1
  151089 37404  8461 1773 343 60 9 1

Links

Crossrefs

Cf. A108447 (row sums), A118342 (column 0).
Cf. A033184, A091370, A091836, A263918.

Programs

  • Maple
    # For the function TreesByArityOfTheRoot_Row(m, n) see A263918.
A263917_row := n -> TreesByArityOfTheRoot_Row(2,n):
seq(A263917_row(n), n=0..9); # Peter Luschny, Oct 31 2015
  • Mathematica
    rows = 9;
f[] = 1; Do[f[x] = 1 + x*f[x]*(f[x]^2 + f[x] - 1) + O[x]^(rows+1) // Normal, {rows+1}];
coes = CoefficientList[f[x]^3/(1 - x*t*f[x]) + O[x]^(rows+1), x];
row[n_] := CoefficientList[coes[[n+1]], t];
Table[row[n], {n, 0, rows}] // Flatten (* Jean-François Alcover, Jul 19 2018 *)

Formula

O.g.f. f^3(x)/(1 - x*t*f(x)), where f(x) = 1 + x + 4*x^2 + 20*x^3 + 113*x^4 + ... satisfies 1 + x*f^3(x)/(1 - x*f(x)) = f(x);
f(x) is the o.g.f. for A108447.
First column o.g.f f(x)^3 is the o.g.f. for A118342.
f(x) - 1 is the g.f. for the row sums of the array.

A263918 Riordan array (f(x)^4, f(x)), where 1 + x*f^4(x)/(1 - x*f(x)) = f(x).

Original entry on oeis.org

1, 4, 1, 26, 5, 1, 192, 35, 6, 1, 1531, 270, 45, 7, 1, 12848, 2215, 362, 56, 8, 1, 111818, 18961, 3054, 461, 68, 9, 1, 1000068, 167455, 26670, 4067, 592, 81, 10, 1, 9135745, 1514590, 239081, 36232, 5274, 732, 95, 11, 1
Offset: 0

Views

Author

Peter Bala, Oct 29 2015

Keywords

Comments

Riordan arrays of the form (f(x)^(m+1), f(x)), where f(x) satisfies 1 + x*f^(m+1)(x)/(1 - x*f(x)) = f(x) include (modulo differences of offset) the Motzkin triangle A091836 (m = -1), the Catalan triangle A033184 (m = 0) and the Schroder triangle A091370 (m = 1). This is the case m = 3. See A263917 for the case m = 2.
The coefficients of the power series solution of the equation 1 + x*f^(m+1)(x)/(1 - x*f(x)) = f(x) appear to be given by [x^0] f(x) = 1 and [x^n] f(x) = 1/n * Sum_{k = 1..n} binomial(n,k)*binomial(n + m*k, k - 1) for n >= 1.
This triangle appears in Novelli et al., Figure 8, p. 24, where a combinatorial interpretation is given in terms of trees.

Examples

			Triangle begins
1,
4, 1,
26, 5, 1,
192, 35, 6, 1,
1531, 270, 45, 7, 1,
12848, 2215, 362, 56, 8, 1,
111818, 18961, 3054, 461, 68, 9, 1,
...
f(x) = 1 + x + 5*x^2 + 32*x^3 + 234*x^4 + 1854*x^5 + 15490*x^6 + 134380*x^7 + 1198944*x^8 + 10931761*x^9 + 101412677*x^10 + 954155059*x^11 + 9083120975*x^12 + ...
f(x)^4 = 1 + 4*x + 26*x^2 + 192*x^3 + 1531*x^4 + 12848*x^5 + 111818*x^6 + 1000068*x^7 + 9135745*x^8 + 84880196*x^9 + 799602464*x^10 + 7619763776*x^11 + 73322247876*x^12 + ...

Links

Crossrefs

Cf. A033184, A091370, A091836, A263917.

Programs

  • Maple
    TreesByArityOfTheRoot_Row := proc(m, row) local c,f,s;
c := N -> hypergeom([1-N, seq((N+j)/m, j=m+1..2*m)],
[2, seq((N+j)/(m-1), j=m+1..2*m-1)], -m^m/(m-1)^(m-1)):
f := n -> 1 + add(simplify(c(i))*x^i,i=1..n):
s := j -> coeff(series(f(j)^(m+1)/(1-x*t*f(j)),x,j+1),x,j):
seq(coeff(s(row),t,j),j=0..row) end:
A263918_row := n -> TreesByArityOfTheRoot_Row(3, n):
seq(A263918_row(n), n=0..8); # Peter Luschny, Oct 31 2015
  • Mathematica
    nmax = 9; For[f = 1; n = 1, n <= nmax, n++, f = 1 + x*f^4/(1 - x*f) + O[x]^n // Normal];
g = f^4/(1 - x*t*f) + O[x]^nmax // Normal;
row[n_] := CoefficientList[Coefficient[g, x, n], t];
Table[row[n], {n, 0, nmax}] // Flatten (* Jean-François Alcover, Dec 03 2017 *)

Formula

O.g.f. f^4(x)/(1 - x*t*f(x)) = 1 + (4 + t)*x + (26 + 5*t + t^2)*x^2 + ..., where f(x) = 1 + x + 5*x^2 + 32*x^3 + 234*x^4 + ... satisfies 1 + x*f^4(x)/(1 - x*f(x)) = f(x);
f(x) - 1 is the g.f. for the row sums of the array.
f(x)^4 is the g.f. for the first column of the array.
Showing 1-3 of 3 results.

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

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