A218473 Number of 3n-length 3-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

1, 1, 7, 61, 591, 6101, 65719, 729933, 8297247, 96044101, 1128138567, 13411861629, 161066465583, 1950996039669, 23808159962839, 292413627476141, 3611870017079871, 44838216520062117, 559127724970143079, 7000374603097246173, 87964883375131331151

Offset

0,3

Links

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

Formula

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 prime p and positive integers n and k. (End)

Maple

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);
# second Maple program
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);

Mathematica

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 *)

Crossrefs

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

Sequence in context: A015572 A066443 A108448 * A098659 A269731 A199686

Adjacent sequences: A218470 A218471 A218472 * A218474 A218475 A218476

Keyword

nonn,easy

Author

Alois P. Heinz, Oct 29 2012

Status

approved