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

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

Author

Peter Bala, Oct 29 2015

Keywords

• nonn
• tabl
• easy

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

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

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.