A196523 G.f. satisfies A(x) = x + x*A(A(A(A(x)))).

1, 1, 4, 28, 262, 2944, 37666, 532276, 8151322, 133562194, 2320621222, 42475263136, 814932467836, 16326188799508, 340479903535258, 7373196169450312, 165453350568966163, 3840489521467649158, 92072430090995120044, 2276807696679096394552

Offset

1,3

Links

Table of n, a(n) for n=1..20.

Formula

G.f.: A(x) = A(A(x))/(1 + A(A(A(A(A(x)))))).

G.f.: A(x) = Series_Reversion[ x/(1 + A(A(A(x)))) ].

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).

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.

Example

G.f.: A(x) = x + x^2 + 4*x^3 + 28*x^4 + 262*x^5 + 2944*x^6 + 37666*x^7 +...
Given g.f. A(x), A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + x*D;
B = A*(1 + x*E);
C = B*(1 + x*F);
D = C*(1 + x*G);
E = D*(1 + x*H); ...
The solution to the variables in the system of equations are:
A=A(x)/x, B=A(A(x))/x, C=A(A(A(x)))/x, D=A(A(A(A(x))))/x, etc.,
where iterations begin:
A(x) = x + x^2 + 4*x^3 + 28*x^4 + 262*x^5 + 2944*x^6 + 37666*x^7 +...
A(A(x)) = x + 2*x^2 + 10*x^3 + 77*x^4 + 760*x^5 + 8846*x^6 + 116140*x^7 +...
A(A(A(x))) = x + 3*x^2 + 18*x^3 + 153*x^4 + 1608*x^5 + 19566*x^6 +...
A(A(A(A(x)))) = x + 4*x^2 + 28*x^3 + 262*x^4 + 2944*x^5 + 37666*x^6 +...
A(A(A(A(A(x))))) = x + 5*x^2 + 40*x^3 + 410*x^4 + 4930*x^5 + 66530*x^6 +...
A(A(A(A(A(A(x)))))) = x + 6*x^2 + 54*x^3 + 603*x^4 + 7752*x^5 + 110484*x^6 +...
ALTERNATE GENERATING METHOD.
The g.f. A(x) equals the sum of products of {3*k}-iterations of A(x):
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) +...
where A_n(x) = A_{n-1}(A(x)) is the n-th iteration of A(x) with A_0(x)=x.
Related expansions.
x*A_3(x) = x^2 + 3*x^3 + 18*x^4 + 153*x^5 + 1608*x^6 + 19566*x^7 +...
x*A_3(x)*A_6(x) = x^3 + 9*x^4 + 90*x^5 + 1026*x^6 + 13059*x^7 +...
x*A_3(x)*A_6(x)*A_9(x) = x^4 + 18*x^5 + 279*x^6 + 4320*x^7 +...
x*A_3(x)*A_6(x)*A_9(x)*A_12(x) = x^5 + 30*x^6 + 675*x^7 +...

Mathematica

Nest[x + x (# /. x -> # /. x -> # /. x -> #) &, O[x], 30][[3]] (* Vladimir Reshetnikov, Aug 08 2019 *)

Prog

(PARI) /* Define the n-th iteration of function F: */
{ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
/* A(x) results from nested iterations of shifted series: */
(PARI) {a(n)=local(A=x); for(i=1, n, A=x+x*ITERATE(4, A, n)); polcoeff(A, n)}
(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)}

Crossrefs

Cf. A030266, A091713.

Sequence in context: A152410 A177403 A360775 * A260775 A292810 A368892

Adjacent sequences: A196520 A196521 A196522 * A196524 A196525 A196526

Keyword

nonn

Author

Paul D. Hanna, Oct 03 2011

Status

approved