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#22 by Paul D. Hanna at Thu Nov 05 18:01:33 EST 2015
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#21 by Paul D. Hanna at Thu Nov 05 18:01:31 EST 2015
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(3) A( x - x^2 - A(x)^3 ) = x - A(x)^3.
(4) B( x - x^3 - B(x)^2 ) = x - B(x)^2.
(35) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) B(x)^(2*n)/n!.
(46) B(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/n!.
(57) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) B(x)^(2*n)/(n!*x) ).
(68) B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/(n!*x) ).
(7) A( x - x^2 - A(x)^3 ) = x - A(x)^3.
(8) B( x - x^3 - B(x)^2 ) = x - B(x)^2.
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| STATUS
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approved
editing
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#20 by Paul D. Hanna at Wed Nov 04 21:28:00 EST 2015
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#19 by Paul D. Hanna at Wed Nov 04 21:27:58 EST 2015
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| LINKS
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Paul D. Hanna, <a href="/A263532/b263532.txt">Table of n, a(n) for n = 1..300</a>
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| STATUS
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approved
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#18 by Paul D. Hanna at Wed Nov 04 21:08:10 EST 2015
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#17 by Paul D. Hanna at Wed Nov 04 21:08:07 EST 2015
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(7) A(B( x - x^32 - BA(x)^2 )) = 3 ) = x - A(x)^3.
(8) B(A( x - x^23 - AB(x)^3 )) = 2 ) = x - B(x)^2.
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| EXAMPLE
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Also,
Also,
where A(B( x - x^3 - B(x)^2 )) = x.
And
where B(A( x - x^2 - A(x)^3 )) = x.
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approved
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#16 by Paul D. Hanna at Wed Nov 04 09:10:39 EST 2015
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#15 by Paul D. Hanna at Wed Nov 04 09:10:37 EST 2015
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| FORMULA
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(7) A(B( x - x^3 - B(x)^2 )) = x.
(8) B(A( x - x^2 - A(x)^3 )) = x.
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approved
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#14 by Paul D. Hanna at Tue Nov 03 19:19:04 EST 2015
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#13 by Paul D. Hanna at Tue Nov 03 19:19:01 EST 2015
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| PROG
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(PARI) {a(n) = my(A=x, B=x); for(i=1, n, A = serreverse(x - B^2 +x*O(x^n)); B= = serreverse(x - A^3); ); polcoeff(A, n)}
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| STATUS
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approved
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