A227372 G.f.: A(x,q) = 1 + x*A(q*x,q) * A(x,q)^2.

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, 210, 196, 196, 174, 156, 121, 85, 59, 42, 27, 14, 9, 4, 2, 1, 429, 792, 840, 882, 842, 796, 749, 627, 480, 382, 289, 216, 157, 101, 67, 46, 27, 14, 9, 4, 2, 1, 1430, 3003

Offset: 0

Author

Paul D. Hanna, Jul 09 2013

Keywords

• nonn
• tabf

Examples

			Triangle begins:
[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, 210, 196, 196, 174, 156, 121, 85, 59, 42, 27, 14, 9, 4, 2, 1];
[429, 792, 840, 882, 842, 796, 749, 627, 480, 382, 289, 216, 157, 101, 67, 46, 27, 14, 9, 4, 2, 1];
[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]; ...
Explicitly, the polynomials in q begin:
1;
1;
2 + q;
5 + 4*q + 2*q^2 + q^3;
14 + 15*q + 10*q^2 + 9*q^3 + 4*q^4 + 2*q^5 + q^6;
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;
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; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1350 (rows 0..20 of triangle, flattened).

Crossrefs

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

Programs

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

Formula

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

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

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

Antidiagonal sums equal A227373.

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