A354019 G.f. A(x) satisfies: A(x)^3 = 36*x + 1/A(x)^3.

1, 6, 18, -288, -1890, 41472, 324324, -7962624, -67343562, 1751777280, 15489019260, -417368899584, -3797625904020, 104791699095552, 972776481568200, -27305722735755264, -257250740550710490, 7314721255213498368, 69699818292739559820

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

(1) A(x)^3 = 18*x + sqrt(1 + 324*x^2).

(2) A(x)^3 = 36*x + 1/A(x)^3.

(3) A(x)^3 = x / Series_Reversion( x/sqrt(1 - 36*x) ).

(4) A(x) = 1/(1 - 36*x/A(x)^3)^(1/6).

(5) A( x/sqrt(1 - 36*x) ) = 1/(1 - 36*x)^(1/6).

(6) A(x)*A(-x) = 1.

(7) A'(x) = 6*A(x) / sqrt(1 + 324*x^2).

(8) A(x) = exp( Integral 6/sqrt(1 + 324*x^2) dx ).

a(n) ~ cos(Pi*(n/2 - 2/3)) * 2^(n + 1/2) * 3^(2*n - 1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, May 15 2022

D-finite with recurrence n*(n-1)*a(n) +36*(3*n-5)*(3*n-7)*a(n-2)=0. - R. J. Mathar, Jan 25 2023

Example

G.f.: A(x) = 1 + 6*x + 18*x^2 - 288*x^3 - 1890*x^4 + 41472*x^5 + 324324*x^6 - 7962624*x^7 - 67343562*x^8 + ...
where A(x)^3 = 36*x + 1/A(x)^3, as seen by comparing the following series:
A(x)^3 = 1 + 18*x + 162*x^2 - 13122*x^4 + 2125764*x^6 - 430467210*x^8 + 97629963228*x^10 + ...
1/A(x)^3 = 1 - 18*x + 162*x^2 - 13122*x^4 + 2125764*x^6 - 430467210*x^8 + ...

Prog

(PARI) my(x='x+O('x^22)); Vec((18*x + sqrt(1 + 324*x^2))^(1/3))
(PARI) {a(n) = my(A = (18*x + sqrt(1 + 324*x^2  +x*O(x^n)) )^(1/3)); polcoeff(A, n)}
for(n=0, 21, print1(a(n), ", "))
(PARI) {a(n) = my(A = (x / serreverse( x/sqrt(1 - 36*x +x*O(x^n)) ))^(1/3)); polcoeff(A, n)}
for(n=0, 21, print1(a(n), ", "))
(PARI) {a(n) = my(A = exp( intformal( 6/sqrt(1 + 324*x^2 +x*O(x^n)) ))); polcoeff(A, n)}
for(n=0, 21, print1(a(n), ", "))

Crossrefs

Sequence in context: A377532 A059944 A052139 * A052682 A214592 A372522

Adjacent sequences: A354016 A354017 A354018 * A354020 A354021 A354022

Keyword

sign

Author

Paul D. Hanna, May 14 2022

Status

approved