A052154 Array read by antidiagonals: a(n,k)= coefficient of z^n of p_k(z), where p_k+1(z)=(p_k(z))^2+z, p_1(z)=z.

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 1, 1, 2, 1, 0, 0, 1, 1, 2, 5, 0, 0, 0, 1, 1, 2, 5, 6, 0, 0, 0, 1, 1, 2, 5, 14, 6, 0, 0, 0, 1, 1, 2, 5, 14, 26, 4, 0, 0, 0, 1, 1, 2, 5, 14, 42, 44, 1, 0, 0, 0, 1, 1, 2, 5, 14, 42, 100, 69, 0, 0, 0, 0

Offset: 1

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 24 2000

Keywords

• nice
• nonn
• tabl
• easy

Comments

a(n,k+1)=a(n,k), n<=k; a(n,n)=A000108. Note that the set {z: limit(p_k(z),k->infinity) not=infinity} of complex numbers defines the Mandelbrot set.

Examples

			p_1(z)=z: coefficient = 1 = a(1,1); p_2(z)=z^2+z: coefficients = 1, 1 = a(1,2), a(2,2); p_3(z)=(z^2+z)^2+z=z+z^2+2z^3+z^4: coefficients = 1,1,2,1 = (1,3), a(2,3), a(3,3), a(4,3); ...
Triangle starts:
1,
1, 0,
1, 1, 0,
1, 1, 0, 0,
1, 1, 2, 0, 0,
1, 1, 2, 1, 0, 0,
1, 1, 2, 5, 0, 0, 0,
1, 1, 2, 5, 6, 0, 0, 0,
1, 1, 2, 5, 14, 6, 0, 0, 0,
1, 1, 2, 5, 14, 26, 4, 0, 0, 0,
1, 1, 2, 5, 14, 42, 44, 1, 0, 0, 0,
1, 1, 2, 5, 14, 42, 100, 69, 0, 0, 0, 0,
...
		

Links

• R. P. Munafo, Mu-Ency - The Encyclopedia of the Mandelbrot Set
• Eric Weisstein's World of Mathematics, Mandelbrot Set
• R. Munafo, Coefficients of Lemniscates for Mandelbrot Set

Crossrefs

Cf. A000108.

Cf. A137560, which gives the same array read by rows. [From Robert Munafo, Dec 12 2009]

Programs

Mathematica

p[1, z_] := z; p[k_, z_] := p[k, z] = p[k-1, z]^2 + z; a[n_, k_] := Coefficient[p[k, z], z, n]; Flatten[ Table[a[n-k, k], {n, 1, 13}, {k, n-1, 1, -1}]] (* Jean-François Alcover, Jun 13 2012 *)

Formula

a(n, k+1)=sum(a(i, k)*a(n-i, k), i=1..n-1) for n=2..2^k, a(1, k)=1, a(n, k+1)=0 for n>2^k.