|
%I #31 Jan 19 2019 04:14:58
%S 1,1,0,1,1,0,1,1,1,0,1,1,4,1,0,1,1,7,19,1,0,1,1,10,61,98,1,0,1,1,13,
%T 127,591,531,1,0,1,1,16,217,1810,6101,2974,1,0,1,1,19,331,4085,27631,
%U 65719,17060,1,0,1,1,22,469,7746,82593,441604,729933,99658,1,0
%N Number A(n,k) of 3n-length k-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; square array A(n,k), n>=0, k>=0, by antidiagonals.
%C In general, column k > 1 is asymptotic to a(n) ~ 3^(3*n+1/2) * (k-1)^(n+1) / (sqrt(Pi) * (2*k-3)^2 * 4^n * n^(3/2)). - _Vaclav Kotesovec_, Aug 31 2014
%H Alois P. Heinz, <a href="/A213027/b213027.txt">Antidiagonals n = 0..140, flattened</a>
%F A(n,k) = 1/n * Sum_{j=0..n-1} C(3*n,j) * (n-j) * (k-1)^j if n>0, k>1; A(0,k) = 1; A(n,k) = k if n>0, k<2.
%F A(n,k) = 1/k * A213028(n,k) if n>0, k>1; else A(n,k) = A213028(n,k).
%e A(0,k) = 1: the empty word.
%e A(n,1) = 1: (aaa)^n.
%e A(2,2) = 4: there are 4 words of length 6 over alphabet {a,b}, 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: aaaaaa, aaabbb, aabbba, abbbaa.
%e A(2,3) = 7: aaaaaa, aaabbb, aaaccc, aabbba, aaccca, abbbaa, acccaa.
%e A(3,2) = 19: aaaaaaaaa, aaaaaabbb, aaaaabbba, aaaabbbaa, aaabaaabb, aaabbaaab, aaabbbaaa, aaabbbbbb, aabaaabba, aabbaaaba, aabbbaaaa, aabbbabbb, aabbbbbba, abaaabbaa, abbaaabaa, abbbaaaaa, abbbaabbb, abbbabbba, abbbbbbaa.
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 4, 7, 10, 13, 16, ...
%e 0, 1, 19, 61, 127, 217, 331, ...
%e 0, 1, 98, 591, 1810, 4085, 7746, ...
%e 0, 1, 531, 6101, 27631, 82593, 195011, ...
%e 0, 1, 2974, 65719, 441604, 1751197, 5153626, ...
%p A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k,
%p 1/n *add(binomial(3*n, j) *(n-j) *(k-1)^j, j=0..n-1))):
%p seq(seq(A(n, d-n), n=0..d), d=0..12);
%t a[0, _] = 1; a[_, k_ /; k < 2] := k; a[n_, k_] := 1/n*Sum[Binomial[3*n, j]*(n-j)*(k-1)^j, {j, 0, n-1}]; Table[a[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Dec 11 2013 *)
%Y Columns k=0-10 give: A000007, A000012, A047099, A218473, A218474, A218475, A218476, A218477, A218478, A218479, A218480.
%Y Rows n=0-3 give: A000012, A057427, A016777(k-1), A127854(k-1).
%Y Main diagonal gives: A218472.
%Y Cf. A183134, A183135, A213028.
%K nonn,tabl
%O 0,13
%A _Alois P. Heinz_, Jun 03 2012
|
