A263918 -id:A263918 - cp's OEIS frontend

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

Peter Bala, Oct 29 2015

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.

			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

J.-C. Novelli and J.-Y. Thibon, Noncommutative Symmetric Functions and Lagrange Inversion, arXiv:math/0512570 [math.CO], 2005-2006.

Cf. A108447 (row sums), A118342 (column 0).

Cf. A033184, A091370, A091836, A263918.

# 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

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 *)

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.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula