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Search: id:A138739
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| A138739 |
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G.f. A(x) satisfies: 3*A(x) = A(A(x)) + 2*x + x^2 with A(0)=0. |
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+0
6
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| 1, 1, 2, 11, 88, 888, 10572, 143214, 2159154, 35702442, 640873656, 12394383780, 256762580460, 5671209169168, 133041670286160, 3304034094162183, 86616702087692256, 2390831825522972392, 69323685702986714272
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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All self-compositions of A(x) may be expressed as a finite sum involving powers of A(x) and x.
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EXAMPLE
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G.f.: A(x) = x + x^2 + 2*x^3 + 11*x^4 + 88*x^5 + 888*x^6 + 10572*x^7 +...
A(A(x)) = x + 2*x^2 + 6*x^3 + 33*x^4 + 264*x^5 + 2664*x^6 + 31716*x^7 +...
so that A(A(x)) + 2*x + x^2 = 3*A(x).
Self-compositions of A=A(x) may be expressed in terms of A and x:
A(A(x)) = 3*A - 2*x - x^2 ;
A(A(A(x))) = (7*A - A^2) - 6*x - 3*x^2 ;
A(A(A(A(x)))) = (15*A - 12*A^2) + (-14 + 12*A)*x +
(-11 + 6*A)*x^2 - 4*x^3 - x^4 ;
A(A(A(A(A(x))))) = (31*A - 83*A^2 + 14*A^3 - A^4) +
(-12*A^2 + 120*A - 30)*x + (-6*A^2 + 60*A - 63)*x^2 - 48*x^3 - 12*x^4 .
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PROGRAM
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(PARI) {a(n)=local(A=x+x^2); if(n<1, 0, for(i=3, n+1, A=A+polcoeff(subst(A, x, A+x*O(x^i)), i)*x^i); polcoeff(A, n))}
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CROSSREFS
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Cf. A138740.
Sequence in context: A036076 A047797 A107096 this_sequence A106961 A099662 A099693
Adjacent sequences: A138736 A138737 A138738 this_sequence A138740 A138741 A138742
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Mar 27 2008
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