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A154757
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G.f. A(x) = Sum_{n>=0} A_n(x)^(n+1) where A_{n+1}(x) = A_n( A(x) ) with A_0(x)=x and A(0)=0.
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1
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1, 1, 3, 14, 83, 574, 4433, 37274, 335751, 3204616, 32159272, 337383121, 3684215369, 41735965645, 489173496890, 5919292080009, 73818803114991, 947369389903293, 12496580900037441, 169248542357479944
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OFFSET
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1,3
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LINKS
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Table of n, a(n) for n=1..20.
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FORMULA
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G.f. A(x) satisfies: A_n(x) = Sum_{k>=0} ( A_{n+k-1}(x) )^(k+1) ; eg.:
A(x) = x + A(x)^2 + A(A(x))^3 + A(A(A(x)))^4 + A(A(A(A(x))))^5 +...;
A(A(x)) = A(x) + A(A(x))^2 + A(A(A(x)))^3 + A(A(A(A(x))))^4 +...;
A(A(A(x))) = A(A(x)) + A(A(A(x)))^2 + A(A(A(A(x))))^3 +...; ...
Series_Reversion(A(x)) = x - x^2 - A(x)^3 - A(A(x))^4 - A(A(A(x)))^5 -...
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EXAMPLE
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G.f.: A(x) = x + x^2 + 3*x^3 + 14*x^4 + 83*x^5 + 574*x^6 +...;
A(A(x)) = x + 2*x^2 + 8*x^3 + 44*x^4 + 292*x^5 + 2201*x^6 +...;
A(A(A(x))) = x + 3*x^2 + 15*x^3 + 96*x^4 + 715*x^5 + 5921*x^6 +...;
A(A(A(A(x)))) = x + 4*x^2 + 24*x^3 + 176*x^4 + 1464*x^5 + 13322*x^6 +...; ...
Series_Reversion(A(x)) = x - x^2 - x^3 - 4*x^4 - 21*x^5 - 133*x^6 - 959*x^7 -...
ILLUSTRATE DEFINITION:
A(x) = x + A(x)^2 + A(A(x))^3 + A(A(A(x)))^4 +... where:
A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 203*x^6 + 1398*x^7 +...;
A(A(x))^3 = x^3 + 6*x^4 + 36*x^5 + 236*x^6 + 1692*x^7 +...;
A(A(A(x)))^4 = x^4 + 12*x^5 + 114*x^6 + 1032*x^7 + 9367*x^8 +...;
A(A(A(A(x))))^5 = x^5 + 20*x^6 + 280*x^7 + 3440*x^8 +...;
A(A(A(A(A(x)))))^6 = x^6 + 30*x^7 + 585*x^8 + 9490*x^9 +...; ...
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PROG
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(PARI) {a(n)=local(A=x, G, S); if(n<1, 0, for(j=1, n, G=x; S=x; for(i=2, n+1, G=subst(A, x, G+x*O(x^n)); S=S+G^i); A=S); polcoeff(A+x*O(x^n), n))}
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CROSSREFS
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Sequence in context: A224776 A220910 A121687 * A074535 A256337 A256330
Adjacent sequences: A154754 A154755 A154756 * A154758 A154759 A154760
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Jan 15 2009
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STATUS
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approved
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