Revision History for A188194
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#5 by Russ Cox at Fri Mar 30 18:37:26 EDT 2012
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| AUTHOR
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_Paul D. Hanna (pauldhanna(AT)juno.com), _, Mar 23 2011
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Discussion
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| Fri Mar 30
| 18:37
| OEIS Server: https://oeis.org/edit/global/213
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#4 by T. D. Noe at Thu Mar 24 02:07:35 EDT 2011
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#3 by Paul D. Hanna at Wed Mar 23 18:13:50 EDT 2011
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| NAME
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G.f. satisfies: A(x) = Sum_{n>=0} log(1 + 2^n*x*A(x)^2)^n/n!.
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| FORMULA
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G.f. A(x) satisfies:
(4) A(x) = F(x*A(x)^2) )) and F(x) = A(x/F(x)^2), )), where F(x) is the g.f. of A188193.
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| EXAMPLE
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G.f.: A(x) = 1 + 2*x + 14*x^2 + 168*x^3 + 3756*x^4 + 261560*x^5 +...
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| PROG
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((PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, log(1+2^m*x*A^2+x*O(x^n))^m/m!)); polcoeff(A, n)}
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| CROSSREFS
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Cf. A014070, A188193.
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#2 by Paul D. Hanna at Wed Mar 23 18:06:57 EDT 2011
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| NAME
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allocated for Paul D. Hanna
G.f. satisfies: A(x) = Sum_{n>=0} log(1 + 2^n*x*A(x)^2)^n/n!.
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| DATA
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1, 2, 14, 168, 3756, 261560, 80733232, 96730287424, 412733638204832, 6222933783425122080, 334514554099356252794912, 64846889330532757107162199040, 45814974387230048629026769270192768
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| OFFSET
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0,2
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| FORMULA
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G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} C(2^n,n)*x^n*A(x)^(2n),
(2) A(x) = sqrt((1/x)*Series_Reversion(x/B(x)^2)),
(3) A(x) = B(x*A(x)^2) and B(x) = A(x/B(x)^2),
where B(x) = Sum_{n>=0} C(2^n,n)*x^n is the g.f. of A014070.
(4) A(x) = F(x*A(x)^2) and F(x) = A(x/F(x)^2), where F(x) is the g.f. of A188193.
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| EXAMPLE
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G.f.: A(x) = 1 + 2*x + 14*x^2 + 168*x^3 + 3756*x^4 + 261560*x^5 +...
which equals the series:
A(x) = 1 + log(1+2*x*A(x)^2) + log(1+4*x*A(x)^2)^2/2! + log(1+8*x*A(x)^2)^3/3! +...
Let B(x) equal the g.f. of A014070, which begins:
B(x) = 1 + 2*x + 6*x^2 + 56*x^3 + 1820*x^4 +...+ C(2^n,n)*x^n +...
then B(x) = A(x/B(x)^2) and A(x) = B(x*A(x)^2), so that:
A(x) = 1 + 2*x*A(x)^2 + 6*x^2*A(x)^4 + 56*x^3*A(x)^6 + 1820*x^4*A(x)^8 +...+ C(2^n,n)*x^n*A(x)^(2n) +...
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| PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, log(1+2^m*x*A^2+x*O(x^n))^m/m!)); polcoeff(A, n)}
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| CROSSREFS
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Cf. A014070, A188193.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Mar 23 2011
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| STATUS
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approved
proposed
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#1 by Paul D. Hanna at Wed Mar 23 17:58:10 EDT 2011
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| NAME
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allocated for Paul D. Hanna
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| KEYWORD
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allocated
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| STATUS
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approved
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