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G.f.: A(x,q) = 1 + x*A(q*x,q) * A(x,q)^2.
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%I #11 Jul 10 2013 00:00:47

%S 1,1,2,1,5,4,2,1,14,15,10,9,4,2,1,42,56,45,43,34,23,14,9,4,2,1,132,

%T 210,196,196,174,156,121,85,59,42,27,14,9,4,2,1,429,792,840,882,842,

%U 796,749,627,480,382,289,216,157,101,67,46,27,14,9,4,2,1,1430,3003

%N G.f.: A(x,q) = 1 + x*A(q*x,q) * A(x,q)^2.

%H Paul D. Hanna, <a href="/A227372/b227372.txt">Table of n, a(n) for n = 0..1350 (rows 0..20 of triangle, flattened).</a>

%F T(n,k) = [x^n*q^k] A(x,q) for k=0..n*(n-1)/2, n>=0.

%F Column 0 is the Catalan numbers (A000108): T(n,0) = C(2*n,n)/(n+1).

%F Row sums equal A001764: Sum_{k=0..n*(n-1)/2} T(n,k) = C(3*n,n)/(2*n+1).

%F Antidiagonal sums equal A227373.

%F Limit of rows, when read in reverse, yields A227377.

%e Triangle begins:

%e [1];

%e [1];

%e [2, 1];

%e [5, 4, 2, 1];

%e [14, 15, 10, 9, 4, 2, 1];

%e [42, 56, 45, 43, 34, 23, 14, 9, 4, 2, 1];

%e [132, 210, 196, 196, 174, 156, 121, 85, 59, 42, 27, 14, 9, 4, 2, 1];

%e [429, 792, 840, 882, 842, 796, 749, 627, 480, 382, 289, 216, 157, 101, 67, 46, 27, 14, 9, 4, 2, 1];

%e [1430, 3003, 3564, 3942, 3990, 3921, 3848, 3681, 3242, 2732, 2267, 1838, 1489, 1189, 909, 671, 494, 345, 252, 173, 109, 71, 46, 27, 14, 9, 4, 2, 1]; ...

%e Explicitly, the polynomials in q begin:

%e 1;

%e 1;

%e 2 + q;

%e 5 + 4*q + 2*q^2 + q^3;

%e 14 + 15*q + 10*q^2 + 9*q^3 + 4*q^4 + 2*q^5 + q^6;

%e 42 + 56*q + 45*q^2 + 43*q^3 + 34*q^4 + 23*q^5 + 14*q^6 + 9*q^7 + 4*q^8 + 2*q^9 + q^10;

%e 132 + 210*q + 196*q^2 + 196*q^3 + 174*q^4 + 156*q^5 + 121*q^6 + 85*q^7 + 59*q^8 + 42*q^9 + 27*q^10 + 14*q^11 + 9*q^12 + 4*q^13 + 2*q^14 + q^15; ...

%o (PARI) {T(n,k)=local(A=1);for(i=1,n,A=1+x*subst(A,x,q*x)*A^2 +x*O(x^n));polcoeff(polcoeff(A,n,x),k,q)}

%o for(n=0,10,for(k=0,n*(n-1)/2,print1(T(n,k),", "));print(""))

%Y Cf. A227373, A227377, A138158, A000108, A001764.

%K nonn,tabf

%O 0,3

%A _Paul D. Hanna_, Jul 09 2013