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A275414 Triangle read by rows: T(n,k) is the number of multisets of k ternary words with a total of n letters.

%I A275414 #20 May 19 2018 02:25:11
%S A275414 3,9,6,27,27,10,81,126,54,15,243,486,297,90,21,729,1836,1380,540,135,
%T A275414 28,2187,6561,5994,2763,855,189,36,6561,23004,24543,13212,4635,1242,
%U A275414 252,45,19683,78732,96723,59130,23490,6996,1701,324,55,59049,265842,368874,253719
%N A275414 Triangle read by rows: T(n,k) is the number of multisets of k ternary words with a total of n letters.
%C A275414 Ternary analog of A209406. Multiset transformation of A000244.
%H A275414 Alois P. Heinz, <a href="/A275414/b275414.txt">Rows n = 1..141, flattened</a>
%H A275414 <a href="/index/Mu#multiplicative_completely">Index entries for triangles generated by the Multiset Transformation</a>
%F A275414 T(n,1) = A000244(n).
%F A275414 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 A275414 G.f.: Product_{j>=1} (1-y*x^j)^(-3^j). - _Alois P. Heinz_, Apr 13 2017
%e A275414       3
%e A275414       9       6
%e A275414      27      27      10
%e A275414      81     126      54      15
%e A275414     243     486     297      90      21
%e A275414     729    1836    1380     540     135      28
%e A275414    2187    6561    5994    2763     855     189      36
%e A275414    6561   23004   24543   13212    4635    1242     252      45
%e A275414   19683   78732   96723   59130   23490    6996    1701     324      55
%e A275414   59049  265842  368874  253719  111609   36828    9846    2232     405      66
%p A275414 b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
%p A275414       `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
%p A275414        binomial(3^i+j-1, j), j=0..min(n/i, p)))))
%p A275414     end:
%p A275414 T:= (n, k)-> b(n$2, k):
%p A275414 seq(seq(T(n, k), k=1..n), n=1..14);  # _Alois P. Heinz_, Apr 13 2017
%t A275414 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 A275414 T[n_, k_] := b[n, n, k];
%t A275414 Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, May 19 2018, after _Alois P. Heinz_ *)
%Y A275414 Cf. A144067 (row sums), A000244 (column 1), A027468 (subdiagonal ?).
%K A275414 nonn,tabl
%O A275414 1,1
%A A275414 _R. J. Mathar_, Jul 27 2016