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G.f.: A(x) = INV(x-x^2 - x^2*INV(x-2*x^2 - x^2*INV(x-4*x^2 - x^2*INV(x-8*x^2 - x^2*INV(x-16*x^2 - ...))))), where INV(F(x)) = series reversion of F(x).
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%I #9 Aug 25 2024 17:00:46

%S 1,1,3,12,59,341,2274,17380,152492,1546370,18253752,251911212,

%T 4077747676,77875414462,1766771227627,47719589665506,1529956518699126,

%U 58172058892474403,2633354766431360249,142248925696032886182,9135446223074433352932,695222497681195469397031

%N G.f.: A(x) = INV(x-x^2 - x^2*INV(x-2*x^2 - x^2*INV(x-4*x^2 - x^2*INV(x-8*x^2 - x^2*INV(x-16*x^2 - ...))))), where INV(F(x)) = series reversion of F(x).

%e G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 59*x^5 + 341*x^6 + 2274*x^7 +...

%e where A(x) results from nested inversions of shifted series:

%e A(x) = Series_Reversion(x-x^2 - x^2*B(x)), where

%e B(x) = x + 2*x^2 + 9*x^3 + 54*x^4 + 392*x^5 + 3358*x^6 + 33768*x^7 +...

%e B(x) = Series_Reversion(x-2*x^2 - x^2*C(x)), where

%e C(x) = x + 4*x^2 + 33*x^3 + 348*x^4 + 4244*x^5 + 58364*x^6 + 911064*x^7 +...

%e C(x) = Series_Reversion(x-4*x^2 - x^2*D(x)), where

%e D(x) = x + 8*x^2 + 129*x^3 + 2616*x^4 + 59972*x^5 + 1497592*x^6 +...

%e D(x) = Series_Reversion(x-8*x^2 - x^2*E(x)), where

%e E(x) = x + 16*x^2 + 513*x^3 + 20592*x^4 + 928004*x^5 + 45007856*x^6 +...

%e E(x) = Series_Reversion(x-16*x^2 - x^2*F(x)), where

%e F(x) = x + 32*x^2 + 2049*x^3 + 164064*x^4 + 14722052*x^5 + 1417021408*x^6 +...

%o (PARI) {a(n)=local(G=x+x^2); for(k=0, n, G=serreverse(x-2^(n-k)*x^2 - x^2*G+x^3*O(x^n))); polcoeff(G, n)}

%Y Cf. A196710.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Oct 05 2011