OFFSET
0,5
COMMENTS
Main diagonal forms A100227. Secondary diagonal is: T(n+1,n) = (n+1)*A001333(n), where A001333 is the numerators of continued fraction convergents to sqrt(2). 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+3*x^2*y^2)/((1-x*y)*(1-2*x*y-x^2*y^2-x*(1-x*y))).
EXAMPLE
Rows begin:
[1],
[1,1],
[1,2,5],
[1,3,9,13],
[1,4,14,28,33],
[1,5,20,50,85,81],
[1,6,27,80,171,246,197],
[1,7,35,119,301,553,693,477],
[1,8,44,168,486,1064,1724,1912,1153],...
where row sums form 3^n-1 for n>0:
3^1-1 = 1+1
3^2-1 = 1+2+5
3^3-1 = 1+3+9+13
3^4-1 = 1+4+14+28+33
3^5-1 = 1+5+20+50+85+81.
PROG
(PARI) T(n, k, m=3)=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 28 2004
STATUS
approved