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G.f.: A(x) = INV(x-x^2 - x^2*INV(x-2*x^2 - x^2*INV(x-3*x^2 - x^2*INV(x-4*x^2 - x^2*INV(x-5*x^2 - ...))))), where INV(F(x)) = series reversion of F(x).
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%I #9 Aug 25 2024 17:00:43

%S 1,1,3,12,59,340,2240,16583,136185,1229220,12119025,129957670,

%T 1510458859,18958901987,255980593558,3703327622812,57208927916093,

%U 941005426734340,16442518936115083,304566619900207724,5967901050385272977,123436972340208737237,2689201938559338920299

%N G.f.: A(x) = INV(x-x^2 - x^2*INV(x-2*x^2 - x^2*INV(x-3*x^2 - x^2*INV(x-4*x^2 - x^2*INV(x-5*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 + 340*x^6 + 2240*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 + 53*x^4 + 366*x^5 + 2849*x^6 + 24591*x^7 +...

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

%e C(x) = x + 3*x^2 + 19*x^3 + 154*x^4 + 1431*x^5 + 14632*x^6 + 161808*x^7 +...

%e C(x) = Series_Reversion(x-3*x^2 - x^2*D(x)), where

%e D(x) = x + 4*x^2 + 33*x^3 + 345*x^4 + 4094*x^5 + 52855*x^6 + 728291*x^7 +...

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

%e E(x) = x + 5*x^2 + 51*x^3 + 656*x^4 + 9531*x^5 + 149804*x^6 + 2495784*x^7 +...

%e E(x) = Series_Reversion(x-5*x^2 - x^2*F(x)), where

%e F(x) = x + 6*x^2 + 73*x^3 + 1117*x^4 + 19254*x^5 + 357925*x^6 + 7026015*x^7 +...

%e F(x) = Series_Reversion(x-6*x^2 - x^2*G(x)), where

%e G(x) = x + 7*x^2 + 99*x^3 + 1758*x^4 + 35111*x^5 + 754864*x^6 + 17098376*x^7 +...

%o (PARI) {a(n)=local(G=x+x^2); for(k=0, n, G=serreverse(x-(n-k+1)*x^2 - x^2*G+x^3*O(x^n))); polcoeff(G, n)}

%Y Cf. A196711.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Oct 05 2011