A340356 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n / (1 - x^(n+1)*A(x)^2).

1, 2, 6, 27, 144, 848, 5294, 34385, 229895, 1571526, 10933068, 77154348, 550955270, 3973757907, 28905779879, 211818655703, 1562190147576, 11586722064844, 86370917023313, 646728926117338, 4862143288139771, 36687265058186722, 277740810853563225, 2108990691307904601

Offset

0,2

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..350

Formula

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

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

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

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

a(n) ~ c * d^n / n^(3/2), where d = 8.1095436670033855687235661331498800514999688916... and c = 0.2942244566611830970157343920557270211951238... - Vaclav Kotesovec, Jan 07 2021

Example

G.f.: A(x) = 1 + 2*x + 6*x^2 + 27*x^3 + 144*x^4 + 848*x^5 + 5294*x^6 + 34385*x^7 + 229895*x^8 + 1571526*x^9 + 10933068*x^10 + ...
where
A(x) = 1/(1 - x*A(x)^2) + x/(1 - x^2*A(x)^2) + x^2/(1 - x^3*A(x)^2) + x^3/(1 - x^4*A(x)^2) + x^4/(1 - x^5*A(x)^2) + ...
also
A(x) = 1/(1 - x) + x*A(x)^2/(1 - x^2) + x^2*A(x)^4/(1 - x^3) + x^3*A(x)^6/(1 - x^4) + x^4*A(x)^8/(1 - x^5) + ...

Mathematica

(* Calculation of constants {d, c}: *) eq = FindRoot[{1/(1 - r*s^2) + (Log[1-r] + QPolyGamma[0, 2 + 2*Log[s]/Log[r], r])/ (r*s^2*Log[r]) == s, (2*QPolyGamma[1, 2 + 2*Log[s]/Log[r], r]) / (r*s^2*Log[r]^2) == 3*s - 2/(-1 + r*s^2)^2}, {r, 1/8}, {s, 2}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[r*s^2*(-1 + r*s^2)*Log[r]^2 * (((-1 + r*s^2)*(1 + s^2*(-1 + (-1 + r)*s*(-1 + r*s^2))) + (-1 + r)*s^2*((-1 + s*(-1 + r*s^2)^2)*Log[r] + (-2 + 3*s*(-1 + r*s^2)^2)*Log[s]) - (-1 + r)*(-1 + r*s^2)^2 * Derivative[0, 0, 1][QPolyGamma][0, 2 + 2*Log[s]/Log[r], r])/(2*Pi*(-1 + r) * (r*s^2*(4 + s*(-9 + r*s*(4 + 9*s*(3 + r*s^2*(-3 + r*s^2)))))* Log[r]^3 - 4*(-1 + r*s^2)^3 * QPolyGamma[2, 2 + 2*Log[s]/Log[r], r])))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Oct 01 2023 *)

Prog

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

Crossrefs

Cf. A340329, A340338, A340355.

Sequence in context: A030967 A030858 A030932 * A363456 A118192 A338180

Adjacent sequences: A340353 A340354 A340355 * A340357 A340358 A340359

Keyword

nonn

Author

Paul D. Hanna, Jan 06 2021

Status

approved