A307999 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * (1 - (-1)^n*A(x))^(n+1), where A(0) = 0.

1, 3, 5, 15, 65, 255, 961, 3759, 15233, 62655, 260097, 1090623, 4616769, 19698687, 84611841, 365570559, 1587755777, 6928284927, 30358910977, 133532161023, 589348292609, 2609230704639, 11584885657601, 51571340750847, 230129898799105, 1029215591587839, 4612514610282497, 20711143725961215, 93164104646180865, 419778524769746943, 1894404146662522881, 8561776644482695167

Offset

1,2

Links

Table of n, a(n) for n=1..32.

Formula

G.f. A = A(x) satisfies

(1) 1 = (1 - A)/(1 - x^2*(1 - A)^2) + x*(1 + A)^2/(1 - x^2*(1 + A)^2).

(2) A = (1 + A)^2*x + ((1 + A)^3 - 4*A)*x^2 - (1 - A^2)^2*x^3 - (1 - A^2)^2*x^4.

(3) 0 = x^3*(1+x)*A^4 - x^2*A^3 - (2*x^4 + 2*x^3 + 3*x^2 + x)*A^2 + (1-x)^2*A - x*(1-x)*(1+x)^2.

Example

G.f.: A(x) = x + 3*x^2 + 5*x^3 + 15*x^4 + 65*x^5 + 255*x^6 + 961*x^7 + 3759*x^8 + 15233*x^9 + 62655*x^10 + 260097*x^11 + 1090623*x^12 + ...
such that
1 = (1 - A(x)) + x*(1 + A(x))^2 + x^2*(1 - A(x))^3 + x^3*(1 + A(x))^4 + x^4*(1 - A(x))^5 + x^5*(1 + A(x))^6 + x^6*(1 - A(x))^7 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, x^m * (1 - (-1)^m*x*Ser(A))^(m+1) ), #A) ); A[n]}
for(n=1, 40, print1(a(n), ", "))

Crossrefs

Sequence in context: A018719 A154107 A305873 * A018771 A214534 A270548

Adjacent sequences: A307996 A307997 A307998 * A308000 A308001 A308002

Keyword

nonn

Author

Paul D. Hanna, May 09 2019

Status

approved