A212027 G.f. satisfies: A(x) = x + x*A(x) * A(A(A(x))) / A(A(x)).

1, 1, 2, 7, 34, 203, 1405, 10902, 92856, 855879, 8451785, 88756905, 985639145, 11522892870, 141307864173, 1812306734024, 24246751794221, 337657449975797, 4884937564807339, 73291058678194555, 1138608805124859321, 18289927572984177863

Offset

1,3

Links

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

Formula

G.f. satisfies: A(x) = x + x*Sum_{n>=1} Product_{k=1..n} A_{2*n-1}(x), where A_{n}(x) denotes the n-th iteration of A(x).

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 203*x^6 + 1405*x^7 +...
Let A_{n}(x) denote the n-th iteration of g.f. A(x), then
A(A(x)) = x + 2*x^2 + 6*x^3 + 25*x^4 + 132*x^5 + 828*x^6 + 5926*x^7 +...
A_3(x)/A(A(x)) = 1 + x + 4*x^2 + 21*x^3 + 133*x^4 + 965*x^5 + 7780*x^6 +...
A_3(x) = x + 3*x^2 + 12*x^3 + 60*x^4 + 356*x^5 + 2417*x^6 + 18315*x^7 +...
A_5(x) = x + 5*x^2 + 30*x^3 + 205*x^4 + 1550*x^5 + 12745*x^6 +...
A_7(x) = x + 7*x^2 + 56*x^3 + 490*x^4 + 4592*x^5 + 45563*x^6 +...
A_9(x) = x + 9*x^2 + 90*x^3 + 963*x^4 + 10842*x^5 + 127311*x^6 +...
The g.f. satisfies the series of products of odd iterations:
A(x) = x + x*A(x) + x*A(x)*A_3(x) + x*A(x)*A_3(x)*A_5(x) + x*A(x)*A_3(x)*A_5(x)*A_7(x) +...
where some of the initial products begin:
x*A(x) = x^2 + x^3 + 2*x^4 + 7*x^5 + 34*x^6 + 203*x^7 + 1405*x^8 +...
x*A(x)*A_3(x) = x^3 + 4*x^4 + 17*x^5 + 85*x^6 + 495*x^7 + 3282*x^8 +...
x*A(x)*A_3(x)*A_5(x) = x^4 + 9*x^5 + 67*x^6 + 495*x^7 + 3800*x^8 +...
x*A(x)*A_3(x)*A_5(x)*A_7(x) = x^5 + 16*x^6 + 186*x^7 + 1958*x^8 +...
x*A(x)*A_3(x)*A_5(x)*A_7(x)*A_9(x) = x^6 + 25*x^7 + 420*x^8 +...

Prog

(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x+x*A*subst(A, x, subst(A, x, A+x*O(x^n)))/subst(A, x, A+x*O(x^n))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A237645 A117399 A145345 * A056543 A357829 A075834

Adjacent sequences: A212024 A212025 A212026 * A212028 A212029 A212030

Keyword

nonn

Author

Paul D. Hanna, Apr 27 2012

Status

approved