0,3

Alois P. Heinz, Table of n, a(n) for n = 0..300

a(n) = (1/n) * Sum_{j=0..n-1} binomial(3*n,j)*(n-j)*2^j for n>0, a(0) = 1.

a(n) ~ 3^(3*n-3/2)/(sqrt(Pi)*2^(n-1)*n^(3/2)). - Vaclav Kotesovec, May 22 2013

G.f. (for n>0): (1/(81*x-3)+2/((3-81*x)*(1-27*x-3*sqrt(3*x*(27*x-2)))^(2/3))). - Vaclav Kotesovec, Jul 06 2013

From Peter Bala, Feb 06 2022: (Start)

The o.g.f. A(x) satisfies the algebraic equation 8*x - 36*x*A(x) + (54*x - 1)*A(x)^2 + (-27*x + 1)*A(x)^3 = 0.

A(x) = (6 - 4*T(2*x))/(2*T(2*x)^2 - 9*T(2*x) + 9), where T(x) = 1 + x*T(x)^3 is the o.g.f. of A001764.

A(x) = 1 + 2*x*B'(2*x)/B(2*x), where B(x) = 2 + x + 2*x^2 + 6*x^3 + 22*x^4 + 91*x^5 + ... is the o.g.f. of A000139.

exp(Sum_{n >= 1} a(n)*x*n/n) = 1 + x + 4*x^2 + 24*x^3 + 176*x^4 + 1456*x^5 + ... is the o.g.f. of A000309, a power series with integral coefficients. It follows that the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. (End)

a:= n-> `if`(n=0, 1, add(binomial(3*n, j)*(n-j)*2^j, j=0..n-1)/n):

seq(a(n), n=0..20);

# Alternative:

a:= proc(n) a(n):= `if`(n<3, [1, 1, 7][n+1], (-81*(3*n-1)*(3*n-5)*a(n-2)

+(81*n^2-81*n+15)*a(n-1))/ ((2*n-1)*n))

end:

seq(a(n), n=0..20);

Flatten[{1, Table[1/n*Sum[Binomial[3*n, j]*(n-j)*2^j, {j, 0, n-1}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 22 2013 *)

Flatten[{1, Table[FullSimplify[SeriesCoefficient[(1/(81*x-3)+2/((3-81*x)*(1-27*x-3*Sqrt[3*x*(27*x-2)])^(2/3))), {x, 0, n}]], {n, 1, 10}]}] (* Vaclav Kotesovec, Jul 06 2013 *)

Column k=3 of A213027. Cf. A000139, A000309, A001764.

Sequence in context: A066443 A108448 A377585 * A098659 A269731 A199686

Adjacent sequences: A218470 A218471 A218472 * A218474 A218475 A218476

nonn,easy

Alois P. Heinz, Oct 29 2012

approved