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A091713 G.f. satisfies A(x) = x + x*A(A(A(x))). 10
1, 1, 3, 15, 99, 781, 7001, 69253, 742071, 8506775, 103411463, 1324477033, 17785238513, 249432247233, 3642471258187, 55246757713367, 868523130653947, 14127076257342933, 237386074703124457, 4115341407421082869, 73516094755096807279, 1351801707136238290351 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Table of n, a(n) for n=1..22.

FORMULA

G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n+2)) for n>0 with F(x,0)=1. - Paul D. Hanna, Apr 16 2007

G.f.: A(x) = G(x)/[1 + G(G(x))] where G(x) = A(A(x)) = g.f. of A141117.

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

G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=0..n} A_{2*k}(x), where A_n(x) denotes the n-th iteration of A(x) with A_0(x)=x. - Paul D. Hanna, Jul 21 2011

EXAMPLE

G.f.: A(x) = x + x^2 + 3*x^3 + 15*x^4 + 99*x^5 + 781*x^6 +...

From Paul D. Hanna, Apr 16 2007:

G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:

A = 1 + xC;

B = A*(1 + xD);

C = B*(1 + xE);

D = C*(1 + xF);

E = D*(1 + xG) ; ...

The solution to the variables in the system of equations are

A=A(x), B=A(A(x)), C=A(A(A(x))), D=A(A(A(A(x)))), etc., where:

A(x) = x + x^2 + 3*x^3 + 15*x^4 + 99*x^5 + 781*x^6 + 7001*x^7 +...

A(A(x)) = x + 2*x^2 + 8*x^3 + 46*x^4 + 330*x^5 + 2756*x^6 +...

A(A(A(x))) = x + 3*x^2 + 15*x^3 + 99*x^4 + 781*x^5 + 7001*x^6 +...

A(A(A(A(x)))) = x + 4*x^2 + 24*x^3 + 180*x^4 + 1564*x^5 +...

A(A(A(A(A(x))))) = x + 5*x^2 + 35*x^3 + 295*x^4 + 2815*x^5 +...

ALTERNATE GENERATING METHOD.

The g.f. A(x) equals the sum of products of even iterations of A(x):

A(x) = x + x*A_2(x) + x*A_2(x)*A_4(x) + x*A_2(x)*A_4(x)*A_6(x) + x*A_2(x)*A_4(x)*A_6(x)*A_8(x) +...+ Product_{k=0..n} A_{2*k}(x) +...

where A_n(x) = A_{n-1}(A(x)) is the n-th iteration of A(x) with A_0(x)=x.

MATHEMATICA

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

PROG

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

(PARI) /* Define the n-th iteration of F: */

{ITERATE(F, n, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}

/* A(x) equals the sum of products of even iterations of A(x): */

{a(n)=local(A=x); for(i=1, n, A=sum(m=0, n-1, prod(k=0, m, ITERATE(A, 2*k, n)+x*O(x^n)))); polcoeff(A, n)}

CROSSREFS

Cf. A002449, A030266, A087949, A088714, A088717, A120971.

Cf. A141117.

Sequence in context: A046635 A208426 A168344 * A156106 A111546 A219359

Adjacent sequences:  A091710 A091711 A091712 * A091714 A091715 A091716

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 31 2004, Jun 04 2008

STATUS

approved

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Last modified August 3 05:55 EDT 2020. Contains 336197 sequences. (Running on oeis4.)