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