%I #20 May 19 2018 02:25:11

%S 3,9,6,27,27,10,81,126,54,15,243,486,297,90,21,729,1836,1380,540,135,

%T 28,2187,6561,5994,2763,855,189,36,6561,23004,24543,13212,4635,1242,

%U 252,45,19683,78732,96723,59130,23490,6996,1701,324,55,59049,265842,368874,253719

%N Triangle read by rows: T(n,k) is the number of multisets of k ternary words with a total of n letters.

%C Ternary analog of A209406. Multiset transformation of A000244.

%H Alois P. Heinz, <a href="/A275414/b275414.txt">Rows n = 1..141, flattened</a>

%H <a href="/index/Mu#multiplicative_completely">Index entries for triangles generated by the Multiset Transformation</a>

%F T(n,1) = A000244(n).

%F T(n,k) = Sum_{c_i*N_i=n,i=1..k} binomial(T(N_i,1)+c_i-1,c_i) for 1<k<=n.

%F G.f.: Product_{j>=1} (1-y*x^j)^(-3^j). - _Alois P. Heinz_, Apr 13 2017

%e 3

%e 9 6

%e 27 27 10

%e 81 126 54 15

%e 243 486 297 90 21

%e 729 1836 1380 540 135 28

%e 2187 6561 5994 2763 855 189 36

%e 6561 23004 24543 13212 4635 1242 252 45

%e 19683 78732 96723 59130 23490 6996 1701 324 55

%e 59049 265842 368874 253719 111609 36828 9846 2232 405 66

%p b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,

%p `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*

%p binomial(3^i+j-1, j), j=0..min(n/i, p)))))

%p end:

%p T:= (n, k)-> b(n$2, k):

%p seq(seq(T(n, k), k=1..n), n=1..14); # _Alois P. Heinz_, Apr 13 2017

%t b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i-1, p - j]*Binomial[3^i + j - 1, j], {j, 0, Min[n/i, p]}]]]];

%t T[n_, k_] := b[n, n, k];

%t Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, May 19 2018, after _Alois P. Heinz_ *)

%Y Cf. A144067 (row sums), A000244 (column 1), A027468 (subdiagonal ?).

%K nonn,tabl

%O 1,1

%A _R. J. Mathar_, Jul 27 2016