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 = 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.
LINKS
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^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.
EXAMPLE
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 + ...
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 *)
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Oct 29 2015
STATUS
approved