A340334 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n*A(x)^n/(1 - x^2*A(x)^(n+1)).

1, 1, 3, 8, 27, 95, 358, 1401, 5667, 23502, 99499, 428464, 1871746, 8277726, 36999434, 166926834, 759343873, 3479755948, 16052353219, 74497506171, 347642785112, 1630507403067, 7683310158670, 36364004107612, 172812273357172, 824434290989728

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

Links

Table of n, a(n) for n=0..25.

Formula

G.f. A(x) satisfies:

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

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

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

Example

G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 27*x^4 + 95*x^5 + 358*x^6 + 1401*x^7 + 5667*x^8 + 23502*x^9 + 99499*x^10 + 428464*x^11 + 1871746*x^12 + ...
where
A(x) = 1/(1 - x^2*A(x)) + x*A(x)/(1 - x^2*A(x)^2) + x^2*A(x)^2/(1 - x^2*A(x)^3) + x^3*A(x)^3/(1 - x^2*A(x)^4) + x^4*A(x)^4/(1 - x^2*A(x)^5) + ...
also
A(x) = 1/(1 - x*A(x)) + x^2*A(x)/(1 - x*A(x)^2) + x^4*A(x)^2/(1 - x*A(x)^3) + x^6*A(x)^3/(1 - x*A(x)^4) + x^8*A(x)^4/(1 - x*A(x)^5) + ...

Prog

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

Crossrefs

Cf. A340333, A340356.

Sequence in context: A148842 A330783 A148843 * A319787 A148844 A145760

Adjacent sequences: A340331 A340332 A340333 * A340335 A340336 A340337

Keyword

nonn

Author

Paul D. Hanna, Jan 11 2021

Status

approved