A340355 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^(2*n)*A(x)^n / (1 - x^(n+1)*A(x)^(n+2)).

1, 1, 4, 15, 73, 373, 2036, 11518, 67108, 399596, 2421477, 14883232, 92561649, 581401130, 3683031612, 23502839520, 150944260610, 974905750378, 6328238976958, 41261578774953, 270119042203681, 1774773646080126, 11699419572947070, 77355905980770122, 512885661648043804

Offset

0,3

Comments

The g.f. A(x) of this sequence is motivated by the following identity:

Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1-q*r^n)) ;

here, p = x^2*A(x), q = x*A(x)^2, and r = x*A(x).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..240

Formula

G.f. A(x) satisfies the following relations.

(1) A(x) = Sum_{n>=0} x^(2*n) * A(x)^n / (1 - x^(n+1)*A(x)^(n+2)).

(2) A(x) = Sum_{n>=0} x^n * A(x)^(2*n) / (1 - x^(n+2)*A(x)^(n+1)).

(3) A(x) = Sum_{n>=0} x^(n^2+3*n) * A(x)^(n^2+3*n) * (1 - x^(2*n+3)*A(x)^(2*n+3)) / ((1 - x^(n+1)*A(x)^(n+2))*(1 - x^(n+2)*A(x)^(n+1))).

a(n) ~ c * d^n / n^(3/2), where d = 7.060918158410189777854181567407... and c = 0.2611318997628883837033125... - Vaclav Kotesovec, Jan 07 2021

Example

G.f.: A(x) =  1 + x + 4*x^2 + 15*x^3 + 73*x^4 + 373*x^5 + 2036*x^6 + 11518*x^7 + 67108*x^8 + 399596*x^9 + 2421477*x^10 + ...
where
A(x) = 1/(1 - x*A(x)^2) + x^2*A(x)/(1 - x^2*A(x)^3) + x^4*A(x)^2/(1 - x^3*A(x)^4) + x^6*A(x)^3/(1 - x^4*A(x)^5) + x^8*A(x)^4/(1 - x^5*A(x)^6) + ...
also
A(x) = 1/(1 - x^2*A(x)) + x*A(x)^2/(1 - x^3*A(x)^2) + x^2*A(x)^4/(1 - x^4*A(x)^3) + x^3*A(x)^6/(1 - x^5*A(x)^4) + x^4*A(x)^8/(1 - x^6*A(x)^5) + ...

Prog

(PARI) {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, x^(2*m)*A^m / (1 - x^(m+1)*A^(m+2) +x*O(x^n)) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, x^m*A^(2*m) / (1 - x^(m+2)*A^(m+1) +x*O(x^n)) )); ; polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A039764 A171005 A303229 * A356009 A307996 A230741

Adjacent sequences: A340352 A340353 A340354 * A340356 A340357 A340358

Keyword

nonn

Author

Paul D. Hanna, Jan 05 2021

Status

approved