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A184509
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G.f.: A(x) = 1 + x*F(x)*G(x) where F(x) = A(x/F(x)) and G(x) = A(x*G(x)).
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3
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1, 1, 2, 5, 17, 78, 423, 2547, 16809, 119633, 904868, 7217525, 60369382, 526911858, 4781722888, 44992996528, 437927234508, 4400711725541, 45584253192633, 486049982786691, 5328493141214993, 59997231748407317, 693194446470892036
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f. A(x), along with F(x) = A(x/F(x)) and G(x) = A(x*G(x)), satisfy:
* A(x/A(x)) = 1 + x*F(x/A(x)) since G(x/A(x)) = A(x);
* A(x*A(x)) = 1 + x*A(x)^2*G(x*A(x)) since F(x*A(x)) = A(x);
* A(x/F(x)^2) = 1 + x*F( x/F(x)^2 )/F(x) since F(x) = G(x/F(x)^2);
* A(x*G(x)^2) = 1 + x*G(x)^3*G( x*G(x)^2 ) since G(x) = F(x*G(x)^2).
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 17*x^4 + 78*x^5 + 423*x^6 +...
The function F(x) = A(x/F(x)) is the g.f. of A184510 and begins:
F(x) = 1 + x + x^2 + x^3 + 4*x^4 + 22*x^5 + 103*x^6 + 565*x^7 + 3650*x^8 +...
The function G(x) = A(x*G(x)) is the g.f. of A184511 and begins:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 58*x^4 + 324*x^5 + 2016*x^6 + 13629*x^7 + 98644*x^8 +...
Related expansions:
A(x*A(x)) = 1 + x + 3*x^2 + 11*x^3 + 47*x^4 + 236*x^5 + 1365*x^6 + 8799*x^7 + 61770*x^8 +...
A(x/A(x)) = 1 + x + x^2 - 2*x^4 - 2*x^5 - 4*x^6 - 55*x^7 - 281*x^8 - 1545*x^9 -...
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PROG
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(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x*serreverse(x/A)/serreverse(x*A)+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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