This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A196523 #22 Dec 01 2024 09:21:28
%S A196523 1,1,4,28,262,2944,37666,532276,8151322,133562194,2320621222,
%T A196523 42475263136,814932467836,16326188799508,340479903535258,
%U A196523 7373196169450312,165453350568966163,3840489521467649158,92072430090995120044,2276807696679096394552
%N A196523 G.f. satisfies A(x) = x + x*A(A(A(A(x)))).
%C A196523 Conjecture: a(n) == 1 (mod 3) for n >= 1. - _Paul D. Hanna_, Dec 01 2024
%H A196523 Paul D. Hanna, <a href="/A196523/b196523.txt">Table of n, a(n) for n = 1..400</a>
%F A196523 G.f.: A(x) = A(A(x))/(1 + A(A(A(A(A(x)))))).
%F A196523 G.f.: A(x) = Series_Reversion[ x/(1 + A(A(A(x)))) ].
%F A196523 G.f.: A(x) = x*F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n+3)) for n>0 with F(x,0)=1; further, x*F(x,n) is the n-th iteration of A(x).
%F A196523 G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=0..n} A_{3*k}(x), where A_n(x) denotes the n-th iteration of A(x) with A_0(x)=x.
%e A196523 G.f.: A(x) = x + x^2 + 4*x^3 + 28*x^4 + 262*x^5 + 2944*x^6 + 37666*x^7 +...
%e A196523 Given g.f. A(x), A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations:
%e A196523 A = 1 + x*D;
%e A196523 B = A*(1 + x*E);
%e A196523 C = B*(1 + x*F);
%e A196523 D = C*(1 + x*G);
%e A196523 E = D*(1 + x*H); ...
%e A196523 The solution to the variables in the system of equations are:
%e A196523 A=A(x)/x, B=A(A(x))/x, C=A(A(A(x)))/x, D=A(A(A(A(x))))/x, etc.,
%e A196523 where iterations begin:
%e A196523 A(x) = x + x^2 + 4*x^3 + 28*x^4 + 262*x^5 + 2944*x^6 + 37666*x^7 +...
%e A196523 A(A(x)) = x + 2*x^2 + 10*x^3 + 77*x^4 + 760*x^5 + 8846*x^6 + 116140*x^7 +...
%e A196523 A(A(A(x))) = x + 3*x^2 + 18*x^3 + 153*x^4 + 1608*x^5 + 19566*x^6 +...
%e A196523 A(A(A(A(x)))) = x + 4*x^2 + 28*x^3 + 262*x^4 + 2944*x^5 + 37666*x^6 +...
%e A196523 A(A(A(A(A(x))))) = x + 5*x^2 + 40*x^3 + 410*x^4 + 4930*x^5 + 66530*x^6 +...
%e A196523 A(A(A(A(A(A(x)))))) = x + 6*x^2 + 54*x^3 + 603*x^4 + 7752*x^5 + 110484*x^6 +...
%e A196523 ALTERNATE GENERATING METHOD.
%e A196523 The g.f. A(x) equals the sum of products of {3*k}-iterations of A(x):
%e A196523 A(x) = x + x*A_3(x) + x*A_3(x)*A_6(x) + x*A_3(x)*A_6(x)*A_9(x) + x*A_3(x)*A_6(x)*A_9(x)*A_12(x) +...+ Product_{k=0..n} A_{3*k}(x) +...
%e A196523 where A_n(x) = A_{n-1}(A(x)) is the n-th iteration of A(x) with A_0(x)=x.
%e A196523 Related expansions.
%e A196523 x*A_3(x) = x^2 + 3*x^3 + 18*x^4 + 153*x^5 + 1608*x^6 + 19566*x^7 +...
%e A196523 x*A_3(x)*A_6(x) = x^3 + 9*x^4 + 90*x^5 + 1026*x^6 + 13059*x^7 +...
%e A196523 x*A_3(x)*A_6(x)*A_9(x) = x^4 + 18*x^5 + 279*x^6 + 4320*x^7 +...
%e A196523 x*A_3(x)*A_6(x)*A_9(x)*A_12(x) = x^5 + 30*x^6 + 675*x^7 +...
%t A196523 Nest[x + x (# /. x -> # /. x -> # /. x -> #) &, O[x], 30][[3]] (* _Vladimir Reshetnikov_, Aug 08 2019 *)
%o A196523 (PARI) /* Define the n-th iteration of function F: */
%o A196523 {ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
%o A196523 /* A(x) results from nested iterations of shifted series: */
%o A196523 (PARI) {a(n)=local(A=x); for(i=1,n, A=x+x*ITERATE(4,A, n)); polcoeff(A, n)}
%o A196523 (PARI) {a(n)=local(A=x); for(i=1,n, A=x+x*sum(m=1,n,prod(k=1,m,ITERATE(3*k,A,n)))); polcoeff(A, n)}
%Y A196523 Cf. A030266, A091713.
%K A196523 nonn
%O A196523 1,3
%A A196523 _Paul D. Hanna_, Oct 03 2011