OFFSET
0,3
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))).
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))).
EXAMPLE
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)).
PROG
(PARI) T(n, k, m=4)=if(n<k || k<0, 0, if(k==0, 1, polcoeff(((1+(m-1)*x+sqrt(1+2*(m-3)*x+(m^2-2*m+5)*x^2+x*O(x^k)))/2)^n, k)))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Nov 29 2004
STATUS
approved