A184506 G.f.: A(x) = 1 + x*G(x)/F(x) where F(x) = A(x/F(x)) and G(x) = A(x*G(x)).

1, 1, 0, 2, -3, 27, -91, 723, -3555, 28338, -174027, 1440582, -10280631, 89422482, -713833016, 6548902473, -57199453969, 553760916426, -5219453249126, 53271930913793, -536862065044303, 5767357558711960, -61733919421613462

Offset

0,4

Links

Table of n, a(n) for n=0..22.

Formula

G.f. A(x), along with F(x) = A(x/F(x)) and G(x) = A(x*G(x)), satisfy:

* A(x/A(x)) = 1 + x/F(x/A(x)) since G(x/A(x)) = A(x);

* A(x*A(x)) = 1 + x*G(x*A(x)) since F(x*A(x)) = A(x);

* A(x/F(x)^2) = 1 + x/[F(x)*F(x/F(x)^2)] since F(x) = G(x/F(x)^2);

* A(x*G(x)^2) = 1 + x * G(x)*G(x*G(x)^2) since G(x) = F(x*G(x)^2).

Example

G.f.: A(x) = 1 + x + 2*x^3 - 3*x^4 + 27*x^5 - 91*x^6 + 723*x^7 -+...
The function F(x) = A(x/F(x)) is the g.f. of A184507 and begins:
F(x) = 1 + x - x^2 + 4*x^3 - 16*x^4 + 86*x^5 - 482*x^6 + 3074*x^7 - 20478*x^8 +-...
The function G(x) = A(x*G(x)) is the g.f. of A184508 and begins:
G(x) = 1 + x + x^2 + 3*x^3 + 6*x^4 + 33*x^5 + 79*x^6 + 661*x^7 + 1564*x^8 +...
Related expansions:
A(x*A(x)) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 18*x^5 + 67*x^6 + 326*x^7 + 1503*x^8 +...
A(x/A(x)) = 1 + x - x^2 + 3*x^3 - 12*x^4 + 59*x^5 - 328*x^6 + 2021*x^7 - 13432*x^8 +-...

Prog

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+1/x*serreverse(x/A)*serreverse(x*A)+x*O(x^n)); polcoeff(A, n)}

Crossrefs

Cf. A184507, A184508, A184509.

Sequence in context: A056722 A181225 A143876 * A126203 A126655 A242520

Adjacent sequences: A184503 A184504 A184505 * A184507 A184508 A184509

Keyword

sign

Author

Paul D. Hanna, Jan 16 2011

Status

approved