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

0,3

Links

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

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

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 *)

Crossrefs

Column k=4 of A213027.

Sequence in context: A079241 A270965 A245923 * A234284 A296379 A183538

Adjacent sequences: A218471 A218472 A218473 * A218475 A218476 A218477

Keyword

nonn

Author

Alois P. Heinz, Oct 29 2012

Status

approved