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A199481
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G.f. A(x) satisfies: 1-x = Sum_{n>=0} (-x)^n * A(x)^[n*phi], where phi = (sqrt(5)+1)/2.
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0
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1, 1, 2, 6, 22, 88, 369, 1600, 7122, 32370, 149612, 701046, 3322671, 15901030, 76729203, 372918345, 1823872039, 8969769017, 44330916829, 220062127756, 1096744496191, 5485585559684, 27526824797373, 138542038343946, 699184051431377, 3537456038149028, 17938918589386868
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OFFSET
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0,3
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COMMENTS
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Compare to g.f. C(x) of the Catalan numbers: 1-x = 1 + Sum_{n>=1} (-x)^n*C(x)^(2*n-1).
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LINKS
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 88*x^5 + 369*x^6 + 1600*x^7 +...
where
1-x = 1 - x*A(x) + x^2*A(x)^3 - x^3*A(x)^4 + x^4*A(x)^6 - x^5*A(x)^8 + x^6*A(x)^9 - x^7*A(x)^11 + x^8*A(x)^12 +...+ (-x)^n*A(x)^A000201(n) +...
Exponents form the lower Wythoff sequence, A000201(n) = [n*phi], which begins:
[0,1,3,4,6,8,9,11,12,14,16,17,19,21,22,24,25,27,29,30,32,33,35,...].
Related expansions:
A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 120*x^4 + 498*x^5 + 2153*x^6 +...;
A(x)^4 = 1 + 4*x + 14*x^2 + 52*x^3 + 209*x^4 + 888*x^5 + 3908*x^6 +...;
A(x)^6 = 1 + 6*x + 27*x^2 + 116*x^3 + 507*x^4 + 2274*x^5 + 10415*x^6 +...;
A(x)^8 = 1 + 8*x + 44*x^2 + 216*x^3 + 1030*x^4 + 4904*x^5 + 23476*x^6 +...; ...
where A(x) = 1 + x*A(x)^3 - x^2*A(x)^4 + x^3*A(x)^6 - x^4*A(x)^8 +-...
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PROG
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(PARI) {a(n)=local(A=[1], phi=(sqrt(5)+1)/2); for(i=1, n, A=concat(A, 0); A[#A]=Vec(sum(m=1, #A, (-x)^m*Ser(A)^floor(m*phi)))[#A]); A[n+1]}
(PARI) {a(n)=local(A=1+x, phi=(sqrt(5)+1)/2); for(i=1, n, A=1+sum(m=1, n, -(-x)^m*A^floor((m+1)*phi))+x*O(x^n)); polcoeff(A, n)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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