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%I #19 Dec 07 2020 10:37:14
%S 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,
%T 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,
%U 1,0,1,2,0,6,0,34,0,1,0,1,1,2,0,1,3,0,9,0,55,1,1,0
%N 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.
%C 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], ... .
%H Alois P. Heinz, <a href="/A246690/b246690.txt">Antidiagonals n = 0..140</a>
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, ...
%e 0, 1, 1, 2, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, ...
%e 0, 1, 0, 3, 1, 2, 0, 1, 1, 0, 4, 1, 0, 0, 3, ...
%e 0, 1, 1, 5, 0, 3, 1, 2, 1, 0, 7, 1, 2, 0, 6, ...
%e 0, 1, 0, 8, 0, 4, 0, 3, 2, 1, 13, 2, 0, 0, 10, ...
%e 0, 1, 1, 13, 1, 6, 0, 4, 2, 0, 24, 3, 3, 1, 18, ...
%e 0, 1, 0, 21, 0, 9, 0, 5, 3, 0, 44, 4, 0, 0, 31, ...
%e 0, 1, 1, 34, 0, 13, 1, 7, 4, 0, 81, 5, 5, 0, 55, ...
%e 0, 1, 0, 55, 1, 19, 0, 10, 5, 0, 149, 6, 0, 0, 96, ...
%e 0, 1, 1, 89, 0, 28, 0, 14, 7, 1, 274, 8, 8, 0, 169, ...
%p b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
%p [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
%p end:
%p f:= proc() local i, l; i, l:=0, [];
%p proc(n) while n>=nops(l)
%p do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
%p end
%p end():
%p g:= proc(n, l) option remember; `if`(n=0, 1,
%p add(`if`(i>n, 0, g(n-i, l)), i=l))
%p end:
%p A:= (n, k)-> g(n, f(k)):
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%t b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i>n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
%t f = Module[{i = 0, l = {}}, Function[n, While[n >= Length[l], l = Join[l, b[i, 1]]; i++]; l[[n + 1]]]];
%t g[n_, l_] := g[n, l] = If[n==0, 1, Sum[If[i>n, 0, g[n - i, l]], {i, l}]];
%t A[n_, k_] := g[n, f[k]];
%t Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y 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.
%Y Main diagonal gives A246691.
%Y Cf. A246688, A246720 (the same for partitions).
%K nonn,tabl
%O 0,18
%A _Alois P. Heinz_, Sep 01 2014
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