A212380 - OEIS

%I #7 Sep 03 2024 14:53:56

%S 1,1,2,6,19,65,231,849,3193,12238,47614,187578,746731,2999264,

%T 12139398,49463204,202729889,835242106,3457178366,14369424882,

%U 59949943825,250967906314,1053891879086,4438207804725,18739249857981,79312365115719,336430161046134

%N G.f. satisfies: A(x) = x^2 + A(A(x))/x.

%F G.f. satisfies: A(x) = ( A(x)^3 + A(A(A(x))) ) / A(A(x)).

%e G.f.: A(x) = x^2 + x^3 + 2*x^4 + 6*x^5 + 19*x^6 + 65*x^7 + 231*x^8 +...

%e such that

%e A(A(x)) = x^4 + 2*x^5 + 6*x^6 + 19*x^7 + 65*x^8 + 231*x^9 +...+ a(n+1)*x^(n+2) +...

%e Related expansions:

%e A(A(A(x))) = x^8 + 4*x^9 + 16*x^10 + 62*x^11 + 243*x^12 + 956*x^13 +...

%e A(x)^3 = x^6 + 3*x^7 + 9*x^8 + 31*x^9 + 111*x^10 + 411*x^11 + 1556*x^12 +...

%e A(x)*A(A(x)) = x^6 + 3*x^7 + 10*x^8 + 35*x^9 + 127*x^10 + 473*x^11 + 1799*x^12 +...

%e where A(x)*A(A(x)) = A(x)^3 + A(A(A(x))).

%o (PARI) {a(n)=local(A=x^2+x^3);for(i=1,n,A=x^2+subst(A,x,A+x*O(x^n))/x);polcoeff(A,n)}

%o for(n=2,40,print1(a(n),", "))

%K nonn

%O 2,3

%A _Paul D. Hanna_, May 11 2012