A218474 Number of 3n-length 4-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, 10, 127, 1810, 27631, 441604, 7293700, 123485914, 2131511455, 37368531010, 663539143015, 11908626395320, 215670579863428, 3936425910379840, 72335601620713432, 1337149262553687658, 24847762997547701695, 463900901255209923310, 8697278488612398979645

Offset: 0

Author

Alois P. Heinz, Oct 29 2012

Keywords

• nonn

Links

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

Crossrefs

Column k=4 of A213027.

Programs

Maple

a:= n-> `if`(n=0, 1, add(binomial(3*n, j)*(n-j)*3^j, j=0..n-1)/n):
seq(a(n), n=0..20);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [1, 1, 10][n+1],
      ((2359*n^3 -5063*n^2 +2898*n -360)*a(n-1)
       -576*(3*n-5)*(7*n-2)*(3*n-4)*a(n-2))/
       (2*(2*n-1)*(7*n-9)*n))
    end:
seq(a(n), n=0..30);

Mathematica

a[n_] := If[n == 0, 1, Sum[Binomial[3n, j] (n - j) 3^j, {j, 0, n - 1}]/n];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 18 2020, after Maple *)

Formula

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

a(n) ~ 3^(4*n+3/2) / (25*sqrt(Pi)*n^(3/2)*4^n). - Vaclav Kotesovec, Jul 16 2014