A385910 Triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) where A(x,y) = A(x^3 + 3*x*y*A(x,y)^3, y) / A(x^2 + 2*x*y*A(x,y)^2, y), read by rows.

1, 0, 1, 0, -1, 3, 0, 1, -8, 12, 0, 1, 7, -49, 55, 0, 0, 9, 56, -296, 273, 0, -2, 9, 14, 498, -1815, 1428, 0, 0, -23, 91, -288, 4181, -11284, 7752, 0, -1, -3, -108, 522, -4487, 33168, -70924, 43263, 0, 1, -23, 82, -579, 3971, -49239, 253590, -449616, 246675, 0, 0, 5, -373, 2419, -6510, 46017, -478291, 1892593, -2869779, 1430715

Offset: 1

Author

Paul D. Hanna, Jul 14 2025

Keywords

• sign
• tabl

Comments

Row sums form the Catalan numbers (A000108), with g.f. C(x) = 1 + x*C(x)^2.

Main diagonal equals A001764, with g.f. D(x) = 1 + x*D(x)^3.

Column 1 equals A385911.

Examples

			G.f. A(x,y) = x + y*x^2 + (3*y^2 - y)*x^3 + (12*y^3 - 8*y^2 + y)*x^4 + (55*y^4 - 49*y^3 + 7*y^2 + y)*x^5 + (273*y^5 - 296*y^4 + 56*y^3 + 9*y^2)*x^6 + (1428*y^6 - 1815*y^5 + 498*y^4 + 14*y^3 + 9*y^2 - 2*y)*x^7 + (7752*y^7 - 11284*y^6 + 4181*y^5 - 288*y^4 + 91*y^3 - 23*y^2)*x^8 + (43263*y^8 - 70924*y^7 + 33168*y^6 - 4487*y^5 + 522*y^4 - 108*y^3 - 3*y^2 - y)*x^9 + (246675*y^9 - 449616*y^8 + 253590*y^7 - 49239*y^6 + 3971*y^5 - 579*y^4 + 82*y^3 - 23*y^2 + y)*x^10 + ...
where A(x,y) = A(x^3 + 3*x*y*A(x,y)^3, y) / A(x^2 + 2*x*y*A(x,y)^2, y).
TRIANGLE.
Triangle of coefficients T(n,k) of x^n*y^k in A(x,y) begins
  1;
  0,  1;
  0, -1,   3;
  0,  1,  -8,   12;
  0,  1,   7,  -49,    55;
  0,  0,   9,   56,  -296,   273;
  0, -2,   9,   14,   498, -1815,   1428;
  0,  0, -23,   91,  -288,  4181, -11284,    7752;
  0, -1,  -3, -108,   522, -4487,  33168,  -70924,    43263;
  0,  1, -23,   82,  -579,  3971, -49239,  253590,  -449616,   246675;
  0,  0,   5, -373,  2419, -6510,  46017, -478291,  1892593, -2869779,   1430715;
  0,  0,  -2,  -65, -3746, 28523, -74367,  554792, -4334344, 13891755, -18418400, 8414640; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..1275 (first 50 rows).

Crossrefs

Cf. A384830 (y=-1), A376226 (y=2), A385911 (column 1), A385912 (a diagonal).

Cf. A000108 (row sums), A001764 (main diagonal).

Programs

PARI

{T(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A] = polcoeff( subst(Ax, x, x^3 + 3*y*x*Ax^3 ) - Ax*subst(Ax, x, x^2 + 2*y*x*Ax^2 ), #A+1)); A[n+1]}
\\ Print the rows of the triangle
my(Rown); for(n=1, 12, Rown = T(n); for(k=0,n-1, print1(polcoef(Rown,k),", "));print(""))

Formula

G.f. A(x,y) = Sum_{n>=1} Sum_{k=0..n} T(n,k) * x^n*y^k satisfies the following formulas.

(1) A(x,y) = A(x^3 + 3*x*y*A(x,y)^3, y) / A(x^2 + 2*x*y*A(x,y)^2, y).

(2) A(x,y=1) = C(x) = C(x^3 + 3*x*C(x)^3) / C(x^2 + 2*x*C(x)^2), where C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108).

(3) T(n+1,n) = A001764(n) for n >= 0, with g.f. D(x) = 1 + x*D(x)^3.