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

%I A263918 #12 Dec 03 2017 17:07:05
%S A263918 1,4,1,26,5,1,192,35,6,1,1531,270,45,7,1,12848,2215,362,56,8,1,111818,
%T A263918 18961,3054,461,68,9,1,1000068,167455,26670,4067,592,81,10,1,9135745,
%U A263918 1514590,239081,36232,5274,732,95,11,1
%N A263918 Riordan array (f(x)^4, f(x)), where 1 + x*f^4(x)/(1 - x*f(x)) = f(x).
%C A263918 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.
%C A263918 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 A263918 This triangle appears in Novelli et al., Figure 8, p. 24, where a combinatorial interpretation is given in terms of trees.
%H A263918 J.-C. Novelli and J.-Y. Thibon, <a href="http://arXiv.org/abs/math.CO/0512570">Noncommutative Symmetric Functions and Lagrange Inversion</a>, arXiv:math/0512570 [math.CO], 2005-2006.
%F A263918 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 A263918 f(x) - 1 is the g.f. for the row sums of the array.
%F A263918 f(x)^4 is the g.f. for the first column of the array.
%e A263918 Triangle begins
%e A263918 1,
%e A263918 4, 1,
%e A263918 26, 5, 1,
%e A263918 192, 35, 6, 1,
%e A263918 1531, 270, 45, 7, 1,
%e A263918 12848, 2215, 362, 56, 8, 1,
%e A263918 111818, 18961, 3054, 461, 68, 9, 1,
%e A263918 ...
%e A263918 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 + ...
%e A263918 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 + ...
%p A263918 TreesByArityOfTheRoot_Row := proc(m, row) local c,f,s;
%p A263918 c := N -> hypergeom([1-N, seq((N+j)/m, j=m+1..2*m)],
%p A263918 [2, seq((N+j)/(m-1), j=m+1..2*m-1)], -m^m/(m-1)^(m-1)):
%p A263918 f := n -> 1 + add(simplify(c(i))*x^i,i=1..n):
%p A263918 s := j -> coeff(series(f(j)^(m+1)/(1-x*t*f(j)),x,j+1),x,j):
%p A263918 seq(coeff(s(row),t,j),j=0..row) end:
%p A263918 A263918_row := n -> TreesByArityOfTheRoot_Row(3, n):
%p A263918 seq(A263918_row(n), n=0..8); # _Peter Luschny_, Oct 31 2015
%t A263918 nmax = 9; For[f = 1; n = 1, n <= nmax, n++, f = 1 + x*f^4/(1 - x*f) + O[x]^n // Normal];
%t A263918 g = f^4/(1 - x*t*f) + O[x]^nmax // Normal;
%t A263918 row[n_] := CoefficientList[Coefficient[g, x, n], t];
%t A263918 Table[row[n], {n, 0, nmax}] // Flatten (* _Jean-François Alcover_, Dec 03 2017 *)
%Y A263918 Cf. A033184, A091370, A091836, A263917.
%K A263918 nonn,tabl,easy
%O A263918 0,2
%A A263918 _Peter Bala_, Oct 29 2015