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A340356 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n / (1 - x^(n+1)*A(x)^2). 8
1, 2, 6, 27, 144, 848, 5294, 34385, 229895, 1571526, 10933068, 77154348, 550955270, 3973757907, 28905779879, 211818655703, 1562190147576, 11586722064844, 86370917023313, 646728926117338, 4862143288139771, 36687265058186722, 277740810853563225, 2108990691307904601 (list; graph; refs; listen; history; text; internal format)
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) + ...

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 * A118192 A338180 A307595

Adjacent sequences:  A340353 A340354 A340355 * A340357 A340358 A340359

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 06 2021

STATUS

approved

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Last modified October 1 19:48 EDT 2022. Contains 357172 sequences. (Running on oeis4.)