%I #10 Aug 25 2024 17:00:38
%S 1,1,3,12,58,323,2026,14125,108472,911203,8326290,82382317,879231033,
%T 10088749986,124101412790,1632187723201,22895274500999,
%U 341738132438907,5415659970194984,90928786402251744,1614244644876588572,30243386104969900766,596915061724923842269
%N G.f.: A(x) = INV(x*(1-x) - x^2*INV(x*(1-x)^2 - x^2*INV(x*(1-x)^3 - x^2*INV(x*(1-x)^4 - x^2*INV(x*(1-x)^5 - ...))))), where INV(F(x)) = series reversion of F(x).
%e G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 58*x^5 + 323*x^6 + 2026*x^7 +...
%e where A(x) results from nested inversions of shifted series:
%e A(x) = Series_Reversion(x*(1-x) - x^2*B(x)), where
%e B(x) = x + 2*x^2 + 8*x^3 + 43*x^4 + 276*x^5 + 2014*x^6 + 16313*x^7 +...
%e B(x) = Series_Reversion(x*(1-x)^2 - x^2*C(x)), where
%e C(x) = x + 3*x^2 + 16*x^3 + 110*x^4 + 885*x^5 + 7992*x^6 + 79339*x^7 +...
%e C(x) = Series_Reversion(x*(1-x)^3 - x^2*D(x)), where
%e D(x) = x + 4*x^2 + 27*x^3 + 229*x^4 + 2235*x^5 + 24181*x^6 + 284809*x^7 +...
%e D(x) = Series_Reversion(x*(1-x)^4 - x^2*E(x)), where
%e E(x) = x + 5*x^2 + 41*x^3 + 416*x^4 + 4801*x^5 + 60825*x^6 + 831773*x^7 +...
%e E(x) = Series_Reversion(x*(1-x)^5 - x^2*F(x)), where
%e F(x) = x + 6*x^2 + 58*x^3 + 687*x^4 + 9183*x^5 + 133784*x^6 + 2089453*x^7 +...
%e F(x) = Series_Reversion(x*(1-x)^6 - x^2*G(x)), where
%e G(x) = x + 7*x^2 + 78*x^3 + 1058*x^4 + 16106*x^5 + 265830*x^6 + 4678877*x^7 +...
%o (PARI) {a(n)=local(G=x+x^2); for(k=0, n, G=serreverse(x*(1-x)^(n-k+1) - x^2*G+x^3*O(x^n))); polcoeff(G, n)}
%Y Cf. A194956, A196709.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Oct 05 2011