A100229 Triangle, read by rows, of the coefficients of [x^k] in G100228(x)^n such that the row sums are 4^n-1 for n>0, where G100228(x) is the g.f. of A100228.

1, 1, 2, 1, 4, 10, 1, 6, 21, 35, 1, 8, 36, 92, 118, 1, 10, 55, 185, 380, 392, 1, 12, 78, 322, 879, 1506, 1297, 1, 14, 105, 511, 1715, 3948, 5803, 4286, 1, 16, 136, 760, 3004, 8536, 17020, 21904, 14158, 1, 18, 171, 1077, 4878, 16344, 40395, 71109, 81387, 46763

Offset: 0

Author

Paul D. Hanna, Nov 29 2004

Keywords

• nonn
• tabl

Comments

The main diagonal forms A100230. Secondary diagonal is T(n+1,n) = (n+1)*A052924(n). More generally, if g.f. F(x) satisfies: m^n-b^n = Sum_{k=0..n} [x^k]F(x)^n, then F(x) also satisfies: (m+z)^n - (b+z)^n + z^n = Sum_{k=0..n} [x^k](F(x)+z*x)^n for all z and F(x)=(1+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2; the triangle formed from powers of F(x) will have the g.f.: G(x,y)=(1-2*x*y+m*x^2*y^2)/((1-x*y)*(1-(m-1)*x*y-x^2*y^2-x*(1-x*y))).

Examples

			Rows begin:
[1],
[1,2],
[1,4,10],
[1,6,21,35],
[1,8,36,92,118],
[1,10,55,185,380,392],
[1,12,78,322,879,1506,1297],
[1,14,105,511,1715,3948,5803,4286],
[1,16,136,760,3004,8536,17020,21904,14158],...
where row sums form 4^n-1 for n>0:
4^1-1 = 1+2 = 3
4^2-1 = 1+4+10 = 15
4^3-1 = 1+6+21+35 = 63
4^4-1 = 1+8+36+92+118 = 255
4^5-1 = 1+10+55+185+380+392 = 1023.
The main diagonal forms A100230 = [1,2,10,35,118,392,1297,...],
where Sum_{n>=1} A100230(n)/n*x^n = log((1-x)/(1-3*x-x^2)).
		

Crossrefs

Cf. A100228, A100230, A052924.

Programs

PARI

T(n,k,m=4)=if(n
				

Formula

G.f.: A(x, y)=(1-2*x*y+4*x^2*y^2)/((1-x*y)*(1-3*x*y-x^2*y^2-x*(1-x*y))).