A216171 G.f. A(x) satisfies: A( x - A(x)^3 ) = x+x^3 - A(x)^3.

1, 1, 3, 21, 181, 1815, 20154, 241665, 3080331, 41302128, 578300961, 8410381731, 126533164302, 1963198249559, 31334634890994, 513482316151767, 8625106767201627, 148306563373055094, 2607509241485053311, 46832500213908831048, 858557964325898228058

Offset

1,3

Links

Table of n, a(n) for n=1..21.

Formula

G.f. A(x) satisfies: A(x) = x + G(x)^3 where G(x - A(x)^3) = x.

Example

 G.f.: A(x) = x + x^3 + 3*x^5 + 21*x^7 + 181*x^9 + 1815*x^11 + 20154*x^13 +...
Related expansions:
A(x)^3 = x^3 + 3*x^5 + 12*x^7 + 82*x^9 + 705*x^11 + 6999*x^13 + 76881*x^15 +...
A(x-A(x)^3) = x - 3*x^5 - 12*x^7 - 82*x^9 - 705*x^11 - 6999*x^13 -...
x+x^3 - A(x)^3 = x - 3*x^5 - 12*x^7 - 82*x^9 - 705*x^11 - 6999*x^13 -...
Let G(x) equal the series reversion of x - A(x)^3:
G(x) = x + x^3 + 6*x^5 + 48*x^7 + 467*x^9 + 5124*x^11 + 61284*x^13 + 783129*x^15 +...
then
G(x)^3 = x^3 + 3*x^5 + 21*x^7 + 181*x^9 + 1815*x^11 + 20154*x^13 +...
A(G(x)) = x + 2*x^3 + 12*x^5 + 105*x^7 + 1096*x^9 + 12816*x^11 +...
A(G(x))^3 = x^3 + 6*x^5 + 48*x^7 + 467*x^9 + 5124*x^11 + 61284*x^13 +...
where A(x) = x + G(x)^3 = G(x) + G(x)^3 - A(G(x))^3.

Prog

 (PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x+serreverse(x-A^3+x*O(x^n))^3); polcoeff(A, n)}
for(n=1, 30, print1(a(2*n-1), ", ")) \* only odd-indexed terms are shown *\

Crossrefs

Cf. A214404.

Sequence in context: A295541 A369941 A192946 * A054879 A333090 A131763

Adjacent sequences: A216168 A216169 A216170 * A216172 A216173 A216174

Keyword

nonn

Author

Paul D. Hanna, Sep 03 2012

Status

approved