A182960 G.f.: exp( Sum_{n>=1} C(6n-1,2n-1)*x^n/n ).

1, 5, 95, 2496, 76063, 2524161, 88534548, 3228482908, 121171012431, 4649906785719, 181614908182551, 7196014051078368, 288537887780406468, 11686156771344086156, 477379538839242423528, 19645977861506470428324

Offset

0,2

Comments

The logarithmic derivative of this sequence is a bisection of the logarithmic derivative of A001764 (ternary trees).

To see this, compare the g.f. of this sequence with g.f. of A001764:

exp(Sum_{n>=1} C(3n-1,n-1)*x^n/n) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 +...

Links

Table of n, a(n) for n=0..15.

Formula

G.f. A(x) satisfies: A(x^2) = F(x)*F(-x) where F(x) = 1 + x*F(x)^3 = Sum_{n>=0} C(3n,n)*x^n/(2n+1) is the g.f. of A001764.

Let G(x) = 2*(1+x)^2/(1-2*x+sqrt(1-8*x)), then g.f. A(x) satisfies:

* A(x) = G(x*A(x)^3) and A(x/G(x)^3) = G(x);

* A(x) = [Series_Reversion( x/G(x)^3 ) / x]^(1/3).

The sequence defined by b(n) := [x^n] A(x)^n begins [1, 5, 215, 10463, 537287, 28435880, 1534398353, 83920389642, ...] and conjecturally satisfies the congruence b(p) == b(1) (mod p^3) for prime p >= 5 (checked up to p = 101). - Peter Bala, Sep 14 2021

From Vaclav Kotesovec, Sep 16 2021: (Start)

Recurrence: 16*n*(2*n + 1)*(4*n - 1)*(4*n + 1)*(27*n - 31)*a(n) = 18*(69984*n^5 - 220320*n^4 + 267705*n^3 - 156510*n^2 + 42021*n - 3680)*a(n-1) - 59049*(n-1)*(2*n - 3)*(3*n - 5)*(3*n - 4)*(27*n - 4)*a(n-2).

a(n) ~ sqrt((sqrt(2) - 1)^(2/3) + (sqrt(2) + 1)^(2/3) - 2) * 3^(6*n + 3/2) / (sqrt(Pi) * n^(3/2) * 2^(4*n + 7/2)). (End)

Example

G.f.: A(x) = 1 + 5*x + 95*x^2 + 2496*x^3 + 76063*x^4 + 2524161*x^5 +...
log(A(x)) = 5*x + 165*x^2 + 6188*x^3 + 245157*x^4 + 10015005*x^5 +...+ A025174(2n)*x^n/n +...
G.f. satisfies: A(x) = G(x*A(x)^3) where G(x) begins:
G(x) = 1 + 5*x + 20*x^2 + 96*x^3 + 528*x^4 + 3136*x^5 + 19584*x^6 +...

Mathematica

f[n_] := Exp[ Sum[ Binomial[6 i - 1, 2 i - 1]*x^i/i, {i, n}]]; CoefficientList[ Series[f@ 15, {x, 0, 15}], x]  (* Robert G. Wilson v, Dec 31 2010 *)

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n, binomial(6*m-1, 2*m-1)*x^m/m)+x*O(x^n)), n)}
(PARI) {a(n)=polcoeff((serreverse(x*(1-2*x+sqrt(1-8*x+x*O(x^n)))^3/(8*(1+x)^6))/x)^(1/3), n)}

Crossrefs

Cf. A182959, A025174, A001764, A060941, A300386 - A300389.

Sequence in context: A359271 A363844 A233077 * A241998 A263394 A336942

Adjacent sequences: A182957 A182958 A182959 * A182961 A182962 A182963

Keyword

nonn

Author

Paul D. Hanna, Dec 31 2010

Status

approved