A303059 G.f. A(x) satisfies: x = Sum_{n>=1} (-1)^(n-1) * x^(2*n-1)*A(x) / (1 + x^(2*n-1)*A(x)).

1, 1, 2, 4, 7, 13, 26, 50, 96, 190, 377, 747, 1494, 3004, 6051, 12237, 24843, 50557, 103143, 210975, 432461, 888173, 1827562, 3766980, 7776620, 16077958, 33286760, 69002906, 143213917, 297573927, 618964149, 1288754681, 2685872873, 5602584099, 11696560369, 24438577665, 51100370596, 106926690324, 223896358139, 469129457585

Offset

0,3

Comments

Note that 1 + 4*Sum_{n>=1} (-1)^n * x^(2*n-1)/(1 + x^(2*n-1)) = theta_4(x)^2, where theta_4(x) = 1 + 2*Sum_{n>=1} (-x)^(n^2) is Jacobi's elliptic theta function.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..900

Example

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 13*x^5 + 26*x^6 + 50*x^7 + 96*x^8 + 190*x^9 + 377*x^10 + 747*x^11 + 1494*x^12 + ...
such that
x = x*A(x)/(1 + x*A(x)) - x^3*A(x)/(1 + x^3*A(x)) + x^5*A(x)/(1 + x^5*A(x)) - x^7*A(x)/(1 + x^7*A(x)) + x^9*A(x)/(1 + x^9*A(x)) -+ ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec( sum(m=1, #A, (-1)^m*x^(2*m-1)*Ser(A)/(1+x^(2*m-1)*Ser(A) )) )[#A] ); A[n+1]}
for(n=0, 60, print1(a(n), ", "))

Crossrefs

Sequence in context: A366109 A017995 A251654 * A099155 A342764 A262267

Adjacent sequences: A303056 A303057 A303058 * A303060 A303061 A303062

Keyword

nonn

Author

Paul D. Hanna, Apr 20 2018

Status

approved