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

%I A263917 #15 Jul 19 2018 07:59:07
%S A263917 1,3,1,15,4,1,85,22,5,1,519,132,30,6,1,3330,837,190,39,7,1,22135,5516,
%T A263917 1250,260,49,8,1,151089,37404,8461,1773,343,60,9,1,1052805,259280,
%U A263917 58550,12324,2422,440,72,10,1,7458236,1829018,412375,87045,17283,3214,552,85,11,1
%N A263917 Riordan array (f(x)^3, f(x)), where 1 + x*f^3(x)/(1 - x*f(x)) = f(x).
%C A263917 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.
%C A263917 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.
%C A263917 This triangle appears in Novelli et al., Figure 8, p. 24, where a combinatorial interpretation is given in terms of trees.
%H A263917 J.-C. Novelli and J.-Y. Thibon, <a href="https://arxiv.org/abs/math/0512570">Noncommutative Symmetric Functions and Lagrange Inversion</a>, arXiv:math/0512570 [math.CO], 2005-2006.
%F A263917 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 A263917 f(x) is the o.g.f. for A108447.
%F A263917 First column o.g.f f(x)^3 is the o.g.f. for A118342.
%F A263917 f(x) - 1 is the g.f. for the row sums of the array.
%e A263917 Triangle begins:
%e A263917        1
%e A263917        3     1
%e A263917       15     4     1
%e A263917       85    22     5    1
%e A263917      519   132    30    6   1
%e A263917     3330   837   190   39   7  1
%e A263917    22135  5516  1250  260  49  8 1
%e A263917   151089 37404  8461 1773 343 60 9 1
%p A263917 # For the function TreesByArityOfTheRoot_Row(m, n) see A263918.
%p A263917 A263917_row := n -> TreesByArityOfTheRoot_Row(2,n):
%p A263917 seq(A263917_row(n), n=0..9); # _Peter Luschny_, Oct 31 2015
%t A263917 rows = 9;
%t A263917 f[_] = 1; Do[f[x_] = 1 + x*f[x]*(f[x]^2 + f[x] - 1) + O[x]^(rows+1) // Normal, {rows+1}];
%t A263917 coes = CoefficientList[f[x]^3/(1 - x*t*f[x]) + O[x]^(rows+1), x];
%t A263917 row[n_] := CoefficientList[coes[[n+1]], t];
%t A263917 Table[row[n], {n, 0, rows}] // Flatten (* _Jean-François Alcover_, Jul 19 2018 *)
%Y A263917 Cf. A108447 (row sums), A118342 (column 0).
%Y A263917 Cf. A033184, A091370, A091836, A263918.
%K A263917 nonn,tabl,easy
%O A263917 0,2
%A A263917 _Peter Bala_, Oct 29 2015