OFFSET
0,14
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
Christian Elsholtz, Clemens Heuberger, Daniel Krenn, Algorithmic counting of nonequivalent compact Huffman codes, arXiv:1901.11343 [math.CO], 2019.
Christian Elsholtz, Clemens Heuberger, Helmut Prodinger, The number of Huffman codes, compact trees, and sums of unit fractions, arXiv:1108.5964 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.
EXAMPLE
A(4,1) = 3: [1/4,1/4,1/4,1/8,1/8], [1/2,1/8,1/8,1/8,1/8], [1/2,1/4,1/8,1/16,1/16].
A(5,2) = 7: [1/9,1/9,1/9,1/9,1/9,1/9,1/9,1/9,1/27,1/27,1/27], [1/3,1/9,1/9,1/9,1/9,1/27,1/27,1/27,1/27,1/27,1/27], [1/3,1/9,1/9,1/9,1/9,1/9,1/27,1/27,1/81,1/81,1/81], [1/3,1/3,1/27,1/27,1/27,1/27,1/27,1/27,1/27,1/27,1/27], [1/3,1/3,1/9,1/27,1/27,1/27,1/27,1/27,1/81,1/81,1/81], [1/3,1/3,1/9,1/9,1/27,1/81,1/81,1/81,1/81,1/81,1/81], [1/3,1/3,1/9,1/9,1/27,1/27,1/81,1/81,1/243,1/243,1/243].
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 2, 2, 2, 2, 2, 2, 2, ...
1, 3, 4, 4, 4, 4, 4, 4, 4, ...
1, 5, 7, 8, 8, 8, 8, 8, 8, ...
1, 9, 13, 15, 16, 16, 16, 16, 16, ...
1, 16, 25, 29, 31, 32, 32, 32, 32, ...
1, 28, 48, 57, 61, 63, 64, 64, 64, ...
MAPLE
b:= proc(n, r, k) option remember;
`if`(n<r, 0, `if`(r=0, `if`(n=0, 1, 0), add(
b(n-j, k*(r-j), k), j=0..min(n, r))))
end:
A:= (n, k)-> `if`(k=0, 1, b(k*n+1, 1, k+1)):
seq(seq(A(n, d-n), n=0..d), d=0..14);
MATHEMATICA
b[n_, r_, k_] := b[n, r, k] = If[n < r, 0, If[r == 0, If[n == 0, 1, 0], Sum[b[n - j, k*(r - j), k], {j, 0, Min[n, r]}]]];
A[n_, k_] := If[k == 0, 1, b[k*n + 1, 1, k + 1]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Nov 11 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Nov 08 2017
STATUS
approved