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A294775 Number A(n,k) of partitions of 1 into exactly k*n+1 powers of 1/(k+1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A294775 #28 Mar 21 2019 16:27:08
%S A294775 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,3,1,1,1,1,2,4,5,1,1,1,1,2,4,7,
%T A294775 9,1,1,1,1,2,4,8,13,16,1,1,1,1,2,4,8,15,25,28,1,1,1,1,2,4,8,16,29,48,
%U A294775 50,1,1,1,1,2,4,8,16,31,57,92,89,1,1,1,1,2,4,8,16,32,61,112,176,159,1
%N A294775 Number A(n,k) of partitions of 1 into exactly k*n+1 powers of 1/(k+1); square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A294775 Alois P. Heinz, <a href="/A294775/b294775.txt">Antidiagonals n = 0..140, flattened</a>
%H A294775 Christian Elsholtz, Clemens Heuberger, Daniel Krenn, <a href="https://arxiv.org/abs/1901.11343">Algorithmic counting of nonequivalent compact Huffman codes</a>, arXiv:1901.11343 [math.CO], 2019.
%H A294775 Christian Elsholtz, Clemens Heuberger, Helmut Prodinger, <a href="https://arxiv.org/abs/1108.5964">The number of Huffman codes, compact trees, and sums of unit fractions</a>, arXiv:1108.5964 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.
%e A294775 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].
%e A294775 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].
%e A294775 Square array A(n,k) begins:
%e A294775   1,  1,  1,  1,  1,  1,  1,  1,  1, ...
%e A294775   1,  1,  1,  1,  1,  1,  1,  1,  1, ...
%e A294775   1,  1,  1,  1,  1,  1,  1,  1,  1, ...
%e A294775   1,  2,  2,  2,  2,  2,  2,  2,  2, ...
%e A294775   1,  3,  4,  4,  4,  4,  4,  4,  4, ...
%e A294775   1,  5,  7,  8,  8,  8,  8,  8,  8, ...
%e A294775   1,  9, 13, 15, 16, 16, 16, 16, 16, ...
%e A294775   1, 16, 25, 29, 31, 32, 32, 32, 32, ...
%e A294775   1, 28, 48, 57, 61, 63, 64, 64, 64, ...
%p A294775 b:= proc(n, r, k) option remember;
%p A294775       `if`(n<r, 0, `if`(r=0, `if`(n=0, 1, 0), add(
%p A294775        b(n-j, k*(r-j), k), j=0..min(n, r))))
%p A294775     end:
%p A294775 A:= (n, k)-> `if`(k=0, 1, b(k*n+1, 1, k+1)):
%p A294775 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A294775 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]}]]];
%t A294775 A[n_, k_] := If[k == 0, 1, b[k*n + 1, 1, k + 1]];
%t A294775 Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Nov 11 2017, after _Alois P. Heinz_ *)
%Y A294775 Columns k=0-10 give (offsets may differ): A000012, A002572, A176485, A176503, A194628, A194629, A194630, A194631, A194632, A194633, A295081.
%Y A294775 Main diagonal gives A011782(n-1) for n>0.
%Y A294775 Cf. A294746.
%K A294775 nonn,tabl
%O A294775 0,14
%A A294775 _Alois P. Heinz_, Nov 08 2017