A340338 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n / (1 - x^(n+1)*A(x)^(n+1)).

1, 2, 4, 11, 35, 121, 444, 1689, 6592, 26258, 106313, 436203, 1809727, 7579202, 31999297, 136050472, 582002281, 2503242025, 10818689627, 46959246659, 204623676444, 894785832949, 3925297799901, 17270153317728, 76187650017660, 336934181461844

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..280

Formula

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

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

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

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

a(n) ~ c * d^n / n^(3/2), where d = 4.6940696906229278555829659... and c = 0.71283897646430285525... - Vaclav Kotesovec, Jan 07 2021

Example

G.f.: A(x) = 1 + 2*x + 4*x^2 + 11*x^3 + 35*x^4 + 121*x^5 + 444*x^6 + 1689*x^7 + 6592*x^8 + 26258*x^9 + 106313*x^10 + ...
where
A(x) = 1/(1 - x*A(x)) + x/(1 - x^2*A(x)^2) + x^2/(1 - x^3*A(x)^3) + x^3/(1 - x^4*A(x)^4) + x^4/(1 - x^5*A(x)^5) + ...
also
A(x) = 1/(1 - x) + x*A(x)/(1 - x^2*A(x)) + x^2*A(x)^2/(1 - x^3*A(x)^2) + x^3*A(x)^3/(1 - x^4*A(x)^3) + x^4*A(x)^4/(1 - x^5*A(x)^4) + ...

Mathematica

(* Calculation of constants {d, c}: *) {1/r, Sqrt[-(1-r)*r*s^2*(-1 + r*s)^2*Log[r*s]^5 * (((1 + ((-1 + r)*(1 + s*(-2 + r - (-1 + r)*r*s) - (-1 + r)*s*(-1 + r*s) * Log[r*s]))/((-1 + r*s)*Log[r]))/ (-1 + r)^2 + (s* Derivative[0, 0, 1][QPolyGamma][0, 1 + Log[r]/Log[r*s], r*s])/Log[r])/ (2* Pi*(r*Log[r*s]^3*(-2 - 2*s*(-2 + r*s)*(1 + (-1 + r)*r*s) + (-1 + r)*s* Log[r*s]*(-5 - 4*r*s*(-2 + r*s) - (-1 + r*s)^2*Log[r*s])) + (-1 + r)*(-1 + r*s)^2*(Log[r]^2* QPolyGamma[2, 1 + Log[r]/Log[r*s], r*s] + r*s*Log[r*s]^2*(Log[r*s]*(2 + Log[r*s])* Derivative[0, 0, 1][QPolyGamma][0, 1 + Log[r]/Log[r*s], r*s] - 2*Log[r]* Derivative[0, 0, 1][QPolyGamma][1, 1 + Log[r]/Log[r*s], r*s] + r*s*Log[r*s]^2* Derivative[0, 0, 2][QPolyGamma][0, 1 + Log[r]/Log[r*s], r*s])))))]} /. FindRoot[{1/(1 - r) + (Log[1 - r*s] + QPolyGamma[0, 1 + Log[r]/Log[r*s], r*s])/(r*Log[r*s]) == s, (1/(-1 + r*s) + Derivative[0, 0, 1][QPolyGamma][0, 1 + Log[r]/Log[r*s], r*s])/ Log[r*s] == 1 + (s - 1/(1 - r))/(s*Log[r*s]) + (Log[r]/(r*s*Log[r*s]^3))* QPolyGamma[1, 1 + Log[r]/Log[r*s], r*s]}, {r, 1/4}, {s, 2}, WorkingPrecision -> 120] (* 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^(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^m*A^m / (1 - x^(m+1)*A^m +x*O(x^n)) )); ; polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A071794 A234006 A285002 * A107378 A186998 A243788

Adjacent sequences: A340335 A340336 A340337 * A340339 A340340 A340341

Keyword

nonn

Author

Paul D. Hanna, Jan 05 2021

Status

approved