A196868 G.f. A(x) satisfies: A(x)^2 + A(-x)^2 = 2 and A(x)^3 - A(-x)^3 = 36*x.

1, 6, -18, 144, -1026, 10368, -91044, 995328, -9630090, 109486080, -1120744188, 13042778112, -138540597588, 1637370298368, -17853248637000, 213325958873088, -2371846639850586, 28573129903177728, -322526246042905740, 3910007249908531200, -44670671340291807228

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

From Paul D. Hanna, Apr 04 2022: (Start)

a(2*n+1) = 6 * 24^n * binomial(3*n+1,n)/(3*n+1) for n >= 0.

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

(1) (A(x)^2 + A(-x)^2)/2 = 1.

(2) (A(x)^3 - A(-x)^3)/2 = 18*x.

(3) (A(x) - A(-x))/2 = 6*Series_Reversion(x - 24*x^3).

(4) (A(x) + A(-x))/2 = sqrt( (1 + A(x)*A(-x))/2 ).

(5) A(x)*A(-x) = 1 - 72*Series_Reversion(x - 24*x^3)^2.

(6) A(x) = B(x) + sqrt(1 - B(x)^2), where B(x) = 6*Series_Reversion(x - 24*x^3).

(End)

Example

G.f.: A(x) = 1 + 6*x - 18*x^2 + 144*x^3 - 1026*x^4 + 10368*x^5 +...
where
A(x)^2 = 1 + 12*x + 72*x^3 + 3240*x^5 + 229392*x^7 + 20083464*x^9 +...
A(x)^3 = 1 + 18*x + 54*x^2 + 1134*x^4 + 63180*x^6 + 4903254*x^8 +...

Prog

(PARI) {a(n)=local(A=[1, 6]); for(k=2, n, A=concat(A, 0); if(k%2==0, A[#A]=-Vec(Ser(A)^2)[#A]/2, A[#A]=-Vec(Ser(A)^3)[#A]/3)); A[n+1]}
(PARI) /* Using series reversion: */
{a(n) = my(A, B = 6*serreverse(x - 24*x^3 +x^2*O(x^n)) );
A = B + sqrt(1 - B^2); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Apr 04 2022

Crossrefs

Cf. A196869, A193618, A193619, A196864, A196865, A196866, A196867.

Sequence in context: A012774 A306656 A027744 * A077531 A214537 A052634

Adjacent sequences: A196865 A196866 A196867 * A196869 A196870 A196871

Keyword

sign

Author

Paul D. Hanna, Oct 06 2011

Status

approved