A195196 - OEIS

%I #9 Aug 25 2024 16:38:01

%S 1,1,5,65,2129,187485,47070033,34738190385,76319639400485,

%T 501572306898501857,9878465224237327196177,

%U 583441900063530059086230973,103363150111621009343497895054497,54932966279471274812844096131622516577,87581770519257906006964347036000601388169669

%N G.f.: A(x) = INV(x - x*INV(x - 3*x*INV(x - 3^2*x*INV(x - 3^3*x*INV(x - 3^4*x*INV(x - ...)))))), where INV(F(x)) = series reversion of F(x).

%e G.f.: A(x) = x + x^2 + 5*x^3 + 65*x^4 + 2129*x^5 + 187485*x^6 +...

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

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

%e B(x) = x + 3*x^2 + 45*x^3 + 1755*x^4 + 172449*x^5 + 45558855*x^6 +...;

%e B(x) = Series_Reversion(x - 3*x*C(x)), where

%e C(x) = x + 9*x^2 + 405*x^3 + 47385*x^4 + 13968369*x^5 + 11070801765*x^6 +...;

%e C(x) = Series_Reversion(x - 9*x*D(x)), where

%e D(x) = x + 27*x^2 + 3645*x^3 + 1279395*x^4 + 1131437889*x^5 + 2690204828895*x^6 +...;

%e D(x) = Series_Reversion(x - 27*x*E(x)), where

%e E(x) = x + 81*x^2 + 32805*x^3 + 34543665*x^4 + 91646469009*x^5 + 653719773421485*x^6 +...;

%e E(x) = Series_Reversion(x - 27*x*F(x)), where

%e F(x) = x + 243*x^2 + 295245*x^3 + 932678955*x^4 + 7423363989729*x^5 + 158853904941420855*x^6 +...; ...

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

%Y Cf. A194956, A195194, A195195.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Sep 11 2011