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, flattened
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, 2, 1, 2, 0, 1, 1, 0, 3, 1, 0, 0, 2, ...
0, 1, 1, 3, 0, 2, 1, 2, 1, 0, 4, 1, 2, 0, 4, ...
0, 1, 0, 3, 0, 2, 0, 2, 1, 1, 5, 2, 0, 0, 4, ...
0, 1, 1, 4, 1, 3, 0, 2, 2, 0, 7, 2, 2, 1, 6, ...
0, 1, 0, 4, 0, 3, 0, 2, 1, 0, 8, 2, 0, 0, 6, ...
0, 1, 1, 5, 0, 3, 1, 3, 2, 0, 10, 2, 3, 0, 9, ...
0, 1, 0, 5, 1, 4, 0, 3, 2, 0, 12, 2, 0, 0, 9, ...
0, 1, 1, 6, 0, 4, 0, 3, 2, 1, 14, 3, 3, 0, 12, ...
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, `if`(l=[], 0,
add(g(n-l[-1]*j, subsop(-1=NULL, l)), j=0..n/l[-1])))
end:
A:= (n, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..16);
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, If[l == {}, 0, Sum[g[n - l[[-1]] j, ReplacePart[l, -1 -> Nothing]], {j, 0, n/l[[-1]]}]]];
A[n_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 16}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 02 2014
STATUS
approved