A216683 G.f. satisfies: A(x) = 1 + x*A(x) / ( A(I*x)*A(-I*x) ).

1, 1, 1, 2, 3, 2, 2, 0, -5, -2, -2, 8, 38, 20, 20, -48, -269, -138, -138, 392, 2194, 1132, 1132, -3344, -19010, -9812, -9812, 30032, 172332, 89000, 89000, -279136, -1613629, -833626, -833626, 2663432, 15485978, 8002172, 8002172, -25938768, -151520246, -78309372, -78309372

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

G.f.: A(x) = 1/(1 - G(x^2)/x) where G(x) = x + G(-x)^2 is the g.f. of A143045.

a(4*n+1) = a(4*n+2) for n>=0.

Let A(x) = A0(x^4) + x*A1(x^4) + x^2*A2(x^2) + x^3*A3(x^4), then

(1) A1(x) = A2(x).

(2) A0(x) = A1(x) + x*A3(x) / (2*A0(x)).

(3) A0(x^4) - x^2*A2(x^4) = A(I*x)*A(-I*x).

Example

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 2*x^6 - 5*x^8 - 2*x^9 +...
Related expansions:
1/A(x) = 1 - x - x^3 + 2*x^5 + 3*x^7 - 10*x^9 - 18*x^11 + 68*x^13 + 131*x^15 - 530*x^17 - 1062*x^19 +...+ -A143045(n)*x^(2*n-1) +...
A(I*x)*A(-I*x) = 1 - x^2 + 3*x^4 - 2*x^6 - 5*x^8 + 2*x^10 + 38*x^12 - 20*x^14 - 269*x^16 + 138*x^18 + 2194*x^20 +...
The 4-sections of g.f. A(x) begin:
A0(x) = 1 + 3*x - 5*x^2 + 38*x^3 - 269*x^4 + 2194*x^5 - 19010*x^6 + 172332*x^7 +...
A1(x) = A2(x) = 1 + 2*x - 2*x^2 + 20*x^3 - 138*x^4 + 1132*x^5 - 9812*x^6 + 89000*x^7 +...
A3(x) = 2 + 8*x^2 - 48*x^3 + 392*x^4 - 3344*x^5 + 30032*x^6 - 279136*x^7 + 2663432*x^8 +...
where
A1(x) + x*A3(x)/(2*A0(x)) = 1 + 3*x - 5*x^2 + 38*x^3 - 269*x^4 + 2194*x^5 +...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A/(subst(A, x, I*x+x*O(x^n))*subst(A, x, -I*x+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A212527, A216681, A143045.

Sequence in context: A086764 A255010 A292371 * A367896 A368123 A368128

Adjacent sequences: A216680 A216681 A216682 * A216684 A216685 A216686

Keyword

sign

Author

Paul D. Hanna, Sep 14 2012

Status

approved