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A052154
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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.
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2
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,13
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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.
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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
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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.
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EXAMPLE
| 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); ...
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CROSSREFS
| Cf. A000108.
Cf. A137560, which gives the same array read by rows. [From Robert Munafo (mrob27(AT)gmail.com), Dec 12 2009]
Sequence in context: A002654 A113652 A106139 * A039977 A197548 A029403
Adjacent sequences: A052151 A052152 A052153 * A052155 A052156 A052157
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KEYWORD
| nice,nonn,tabl,easy
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AUTHOR
| Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 24 2000
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