A306064 G.f. A(x) satisfies: A(x) = x + x*A(x + A(x)).

1, 2, 10, 106, 2130, 79442, 5581914, 754371386, 199193444258, 103781218984098, 107308976428238250, 220948088846408617994, 907652841888542054277618, 7447285848965361047618906866, 122120561639979483596993367427066, 4003478037366868501269046319075211994, 262435985468992467742766325045697308601154, 34402470626949826173994511431148770576433889602

Offset

1,2

Links

Table of n, a(n) for n=1..18.

Formula

G.f. A(x) satisfies:

(1) A(x) = x + x*A(x + A(x)).

(2) A(x) = Sum_{n>=0} Product_{k=0..n} B^k(x), where B(x) = x + A(x) and B^n(x) denotes the n-th iteration of B(x).

Example

G.f.: A(x) = x + 2*x^2 + 10*x^3 + 106*x^4 + 2130*x^5 + 79442*x^6 + 5581914*x^7 + 754371386*x^8 + 199193444258*x^9 + 103781218984098*x^10 + ...
such that A(x) = x + x*A(x + A(x)).
RELATED SERIES.
Define B(x) = x + A(x), then
A(x) = x + x*B(x) + x*B(x)*B(B(x)) + x*B(x)*B(B(x))*B(B(B(x))) + x*B(x)*B(B(x))*B(B(B(x)))*B(B(B(B(x)))) + ...
where
B(x) = 2*(x + x^2 + 5*x^3 + 53*x^4 + 1065*x^5 + 39721*x^6 + ...);
B(B(x)) = 4*(x + 3*x^2 + 29*x^3 + 559*x^4 + 20393*x^5 + 1415339*x^6 + ...);
B(B(B(x))) = 8*(x + 7*x^2 + 133*x^3 + 4939*x^4 + 348025*x^5 + ...);
B(B(B(B(x)))) = 16*(x + 15*x^2 + 565*x^3 + 41315*x^4 + 5738713*x^5 + ...);
B(B(B(B(B(x))))) = 32*(x + 31*x^2 + 2325*x^3 + 337683*x^4 + 93186713*x^5 + ...);
etc.

Prog

(PARI) {a(n) = my(A=x); for(i=1, n, A = x + x*subst(A, x, x + A + x*O(x^n))); polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Sequence in context: A049538 A217901 A127728 * A355210 A185396 A003222

Adjacent sequences: A306061 A306062 A306063 * A306065 A306066 A306067

Keyword

nonn

Author

Paul D. Hanna, Jun 28 2018

Status

approved