A246690 Number A(n,k) of compositions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 1, 0, 1, 1, 5, 0, 1, 0, 1, 1, 0, 2, 0, 8, 1, 1, 0, 1, 0, 1, 0, 3, 0, 13, 0, 1, 0, 1, 0, 1, 1, 1, 4, 1, 21, 1, 1, 0, 1, 1, 0, 1, 2, 0, 6, 0, 34, 0, 1, 0, 1, 1, 2, 0, 1, 3, 0, 9, 0, 55, 1, 1, 0

Offset

0,18

Comments

The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .

Links

Alois P. Heinz, Antidiagonals n = 0..140

Example

Square array A(n,k) begins:
  1, 1, 1,  1, 1,  1, 1,  1, 1, 1,   1, 1, 1, 1,   1, ...
  0, 1, 0,  1, 0,  1, 0,  1, 0, 0,   1, 1, 0, 0,   1, ...
  0, 1, 1,  2, 0,  1, 0,  1, 1, 0,   2, 1, 1, 0,   2, ...
  0, 1, 0,  3, 1,  2, 0,  1, 1, 0,   4, 1, 0, 0,   3, ...
  0, 1, 1,  5, 0,  3, 1,  2, 1, 0,   7, 1, 2, 0,   6, ...
  0, 1, 0,  8, 0,  4, 0,  3, 2, 1,  13, 2, 0, 0,  10, ...
  0, 1, 1, 13, 1,  6, 0,  4, 2, 0,  24, 3, 3, 1,  18, ...
  0, 1, 0, 21, 0,  9, 0,  5, 3, 0,  44, 4, 0, 0,  31, ...
  0, 1, 1, 34, 0, 13, 1,  7, 4, 0,  81, 5, 5, 0,  55, ...
  0, 1, 0, 55, 1, 19, 0, 10, 5, 0, 149, 6, 0, 0,  96, ...
  0, 1, 1, 89, 0, 28, 0, 14, 7, 1, 274, 8, 8, 0, 169, ...

Maple

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
f:= proc() local i, l; i, l:=0, [];
      proc(n) while n>=nops(l)
        do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
      end
    end():
g:= proc(n, l) option remember; `if`(n=0, 1,
      add(`if`(i>n, 0, g(n-i, l)), i=l))
    end:
A:= (n, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i>n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[n >= Length[l], l = Join[l, b[i, 1]]; i++]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n==0, 1, Sum[If[i>n, 0, g[n - i, l]], {i, l}]];
A[n_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Crossrefs

Columns k=0-21, 23, 25-28 give: A000007, A000012, A059841, A000045(n+1), A079978, A000930, A121262, A003269(n+1), A182097, A079998, A000073(n+2), A003520, A079977, A079979, A060945, A005708, A001687(n+1), A017817, A082784, A079971, A006498, A005709, A052920, A120400, A060961, A005710, A013979.

Main diagonal gives A246691.

Cf. A246688, A246720 (the same for partitions).

Sequence in context: A214157 A246720 A343030 * A317748 A090465 A364357

Adjacent sequences: A246687 A246688 A246689 * A246691 A246692 A246693

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 01 2014

Status

approved