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

3, 9, 6, 27, 27, 10, 81, 126, 54, 15, 243, 486, 297, 90, 21, 729, 1836, 1380, 540, 135, 28, 2187, 6561, 5994, 2763, 855, 189, 36, 6561, 23004, 24543, 13212, 4635, 1242, 252, 45, 19683, 78732, 96723, 59130, 23490, 6996, 1701, 324, 55, 59049, 265842, 368874, 253719

Offset

1,1

Comments

Ternary analog of A209406. Multiset transformation of A000244.

Links

Alois P. Heinz, Rows n = 1..141, flattened

Index entries for triangles generated by the Multiset Transformation

Formula

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

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.

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

Example

      3
      9       6
     27      27      10
     81     126      54      15
    243     486     297      90      21
    729    1836    1380     540     135      28
   2187    6561    5994    2763     855     189      36
   6561   23004   24543   13212    4635    1242     252      45
  19683   78732   96723   59130   23490    6996    1701     324      55
  59049  265842  368874  253719  111609   36828    9846    2232     405      66

Maple

b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
       binomial(3^i+j-1, j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Apr 13 2017

Mathematica

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[n_, k_] := b[n, n, k];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 19 2018, after Alois P. Heinz *)

Crossrefs

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

Sequence in context: A223918 A224190 A223815 * A223309 A179483 A346108

Adjacent sequences: A275411 A275412 A275413 * A275415 A275416 A275417

Keyword

nonn,tabl

Author

R. J. Mathar, Jul 27 2016

Status

approved