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

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

0,2

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.

Links

Table of n, a(n) for n=0..54.

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

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.

Example

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

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 *)

Crossrefs

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

Cf. A033184, A091370, A091836, A263918.

Sequence in context: A290030 A072479 A264772 * A324428 A131440 A269950

Adjacent sequences: A263914 A263915 A263916 * A263918 A263919 A263920

Keyword

nonn,tabl,easy

Author

Peter Bala, Oct 29 2015

Status

approved