A171212
G.f.: A(x) satisfies A(x) = x + x*A(A(2*x)).
Original entry on oeis.org
0, 1, 2, 16, 320, 12928, 985088, 140861440, 38451150848, 20403322617856, 21307854867660800, 44110759073910095872, 181739941085108158595072, 1493546441998961207249207296, 24512116566896662943648857456640
Offset: 0
G.f.: A(x) = x + 2*x^2 + 16*x^3 + 320*x^4 + 12928*x^5 +...
A(A(x)) = x + 4*x^2 + 40*x^3 + 808*x^4 + 30784*x^5 + 2200960*x^6 +...+ a(n)*x^n/2^(n-1) +...
As a formal series involving products of iterations of the g.f.,
A(x) = x + x*A(2x) + x*A(2x)*A(2A(2x)) + x*A(2x)*A(2A(2x))*A(2A(2A(2x))) +...
which, upon replacing x with A(2x), yields:
A(A(2x)) = A(2x) + A(2x)*A(2A(2x)) + A(2x)*A(2A(2x))*A(2A(2A(2x))) +...
thus A(x) = x + x*A(A(2x)).
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{a(n,q=2)=local(A=x+x^2);for(i=1,n,A=x+x*subst(A,x,subst(A,x,q*x+O(x^n))));polcoeff(A,n)}
A171214
G.f. A(x) satisfies A(x) = x + x*A(A(x/3)) = Sum_{n>=1} a(n)*x^n/3^(n*(n+1)/2).
Original entry on oeis.org
1, 1, 2, 10, 137, 5296, 588365, 190088818, 179954321171, 501722122937995, 4134242130461174144, 100943613343624534183723, 7317423203727305175501741434, 1577227642328692213664066391691150
Offset: 1
G.f.: A(x) = x + x^2/3 + 2*x^3/3^3 + 10*x^4/3^6 + 137*x^5/3^10 + 5296*x^6/3^15 +...+ a(n)*x^n/3^(n(n-1)/2) +...
A(x) = x + x*A(x/3) + x*A(x/3)*A(A(x/3)/3) + x*A(x/3)*A(A(x/3)/3)*A(A(A(x/3)/3)/3) +...
A(A(x)) = x + 2*x^2/3 + 10*x^3/3^3 + 137*x^4/3^6 + 5296*x^5/3^10 +...
SUMS OF SERIES at certain arguments.
A(1) = 1.423879975541542344910599787693637973194...
A(1/3) = 0.373293286580877833612329400906044642790...
A(A(1/3)) = A(1) - 1 = 0.42387997554...
A(A(1)) = 2.387414460111728675082753594461076041830...
A(3) = 3 + 3*A(A(1)) = 10.16224338033518602524826...
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{a(n)=local(A=x+x^2);for(i=1,n,A=x+x*subst(A,x,subst(A,x,x/3+O(x^n))));3^(n*(n-1)/2)*polcoeff(A,n)}
Showing 1-2 of 2 results.
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