A276368 G.f. A(x) satisfies: A(x - 3*x^3) = 1/(1 - 3*x).

1, 3, 9, 36, 135, 567, 2268, 9720, 40095, 173745, 729729, 3184272, 13533156, 59337684, 254304360, 1118939184, 4825425231, 21288640725, 92250776475, 407845538100, 1774128090735, 7856852973255, 34284449337840, 152044079672160, 665192848565700, 2953456247631708, 12949769701154412, 57554532005130720, 252828837022538520, 1124652412962326520, 4948470617034236688, 22028675650023376224

Offset

0,2

Comments

Compare with the o.g.f. G(x) = (1 - sqrt(1 - 8*x))/(4*x) of A151374, which satisfies G(x - 2*x^2) = 1/(1 - 2*x). - Peter Bala, Oct 29 2017

References

R. L. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics (1989), Addison-Wesley, Reading, MA. Sections 5.4 & 7.5.

Links

Table of n, a(n) for n=0..31.

N. D. Elkies, The power series for the inverse function of y(1-y)^t

Formula

G.f. A(x) satisfies:

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

(2) A(x) = 1/(1 - F(x)^2) where F(x) = sqrt(3) * G(sqrt(3)*x) where G(x) = x + G(x)^3 is the g.f. of A001764.

(3) (A(x) - A(-x)) / (A(x) + A(-x)) = sqrt(3) * G(sqrt(3)*x) such that G(x) = x + G(x)^3 is the g.f. of A001764.

(4) A'(x) = 3*A(x)^3 * A(-x).

(5) A(x) = exp( Integral 3*A(x)^2 * A(-x) dx ).

From Peter Bala, Oct 29 2017: (Start)

a(2*n) = 3^n*binomial(3*n, n).

a(2*n+1) = 3^(n+1)*binomial(3*n+1, n).

The proof uses the properties of Lambert's generalized binomial series B_t(z) at t = 3 - see Graham et al., sections 5.4 and 7.5 and also Elkies. (End)

D-finite with recurrence: 4*n*(n-1)*a(n) +18*(-n+1)*a(n-1) -9*(3*n-4)*(3*n-5)*a(n-2)=0. - R. J. Mathar, Jan 27 2020

Example

G.f.: A(x) = 1 + 3*x + 9*x^2 + 36*x^3 + 135*x^4 + 567*x^5 + 2268*x^6 + 9720*x^7 + 40095*x^8 + 173745*x^9 +...
such that A(x - 3*x^3) = 1/(1 - 3*x).
RELATED SERIES.
A(x) * A(-x) = 1 + 9*x^2 + 135*x^4 + 2268*x^6 + 40095*x^8 + 729729*x^10 + 13533156*x^12 + 254304360*x^14 + 4825425231*x^16 +...
which equals (A(x) + A(-x))/2.
(A(x) - A(-x))/2 = 3*x + 36*x^3 + 567*x^5 + 9720*x^7 + 173745*x^9 + 3184272*x^11 + 59337684*x^13 + 1118939184*x^15 + 21288640725*x^17 +...
(A(x) - A(-x)) / (A(x) + A(-x)) = sqrt(3) * G(sqrt(3)*x) where
G(x) = x + x^3 + 3*x^5 + 12*x^7 + 55*x^9 + 273*x^11 + 1428*x^13 + 7752*x^15 +...+ binomial(3*n,n)/(2*n+1)*x^(2*n+1) +...
such that G(x) = x + G(x)^3 is the g.f. of A001764.

Mathematica

nmax = 31; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x - 3 x^3] - 1/(1 - 3 x) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)

Prog

(PARI) {a(n) = my(A=1); A = 1/(1 - 3*serreverse(x - 3*x^3 +x^2*O(x^n))); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n, A = exp(intformal( 3*A^2*subst(A, x, -x) +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))

Crossrefs

Cf. A001764, A151374.

Sequence in context: A183495 A185162 A106435 * A058540 A350451 A245888

Adjacent sequences: A276365 A276366 A276367 * A276369 A276370 A276371

Keyword

nonn,easy

Author

Paul D. Hanna, Sep 02 2016

Status

approved