A354019 - OEIS

%I #12 Jan 25 2023 09:59:12

%S 1,6,18,-288,-1890,41472,324324,-7962624,-67343562,1751777280,

%T 15489019260,-417368899584,-3797625904020,104791699095552,

%U 972776481568200,-27305722735755264,-257250740550710490,7314721255213498368,69699818292739559820

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

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

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

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

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

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

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

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

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

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

%F 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

%F 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

%e 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 + ...

%e where A(x)^3 = 36*x + 1/A(x)^3, as seen by comparing the following series:

%e A(x)^3 = 1 + 18*x + 162*x^2 - 13122*x^4 + 2125764*x^6 - 430467210*x^8 + 97629963228*x^10 + ...

%e 1/A(x)^3 = 1 - 18*x + 162*x^2 - 13122*x^4 + 2125764*x^6 - 430467210*x^8 + ...

%o (PARI) my(x='x+O('x^22)); Vec((18*x + sqrt(1 + 324*x^2))^(1/3))

%o (PARI) {a(n) = my(A = (18*x + sqrt(1 + 324*x^2  +x*O(x^n)) )^(1/3)); polcoeff(A,n)}

%o for(n=0,21, print1(a(n),", "))

%o (PARI) {a(n) = my(A = (x / serreverse( x/sqrt(1 - 36*x +x*O(x^n)) ))^(1/3)); polcoeff(A,n)}

%o for(n=0,21, print1(a(n),", "))

%o (PARI) {a(n) = my(A = exp( intformal( 6/sqrt(1 + 324*x^2 +x*O(x^n)) ))); polcoeff(A,n)}

%o for(n=0,21, print1(a(n),", "))

%K sign

%O 0,2

%A _Paul D. Hanna_, May 14 2022