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%I #36 Jun 06 2021 09:00:26
%S 0,1,1,1,1,1,1,2,2,2,1,2,3,4,3,1,2,4,6,7,5,1,2,4,7,11,12,8,1,2,4,8,14,
%T 21,21,13,1,2,4,8,15,27,39,37,21,1,2,4,8,16,30,52,73,65,34,1,2,4,8,16,
%U 31,59,101,136,114,55,1,2,4,8,16,32,62,116,195,254,200,89
%N Rectangular array read by antidiagonals: T(n,k) is the number of compositions (ordered partitions) of n in which no part (summand) is equal to k.
%H Alois P. Heinz, <a href="/A179647/b179647.txt">Antidiagonals n = 1..141, flattened</a>
%F O.g.f. for column k: B(A(x)) where A(x) = x/(1-x)-x^k and B(x) = 1/(1-x).
%e T(4,2) = 4 because there are four compositions of 4 with no part equal to two: 4, 3+1, 1+3, 1+1+1+1.
%e Array starts as:
%e 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, ...
%e 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, ...
%e 2, 4, 6, 7, 8, 8, 8, 8, 8, 8, ...
%e 3, 7, 11, 14, 15, 16, 16, 16, 16, 16, ...
%e 5, 12, 21, 27, 30, 31, 32, 32, 32, 32, ...
%e 8, 21, 39, 52, 59, 62, 63, 64, 64, 64, ...
%e 13, 37, 73, 101, 116, 123, 126, 127, 128, 128, ...
%e 21, 65, 136, 195, 228, 244, 251, 254, 255, 256, ...
%e 34, 114, 254, 377, 449, 484, 500, 507, 510, 511, ...
%p T:= proc(n, k) option remember;
%p `if`(n=0, 1, add(`if`(j=k, 0, T(n-j, k)), j=1..n))
%p end:
%p seq(seq(T(n, d+1-n), n=1..d), d=1..12); # _Alois P. Heinz_, Oct 23 2011
%t Transpose[Table[Table[Length[Select[Level[Map[Permutations, IntegerPartitions[n]], {2}], FreeQ[#, i] &]], {n, 1, 10}], {i, 1, 10}]] // Grid
%t (* second program: *)
%t a = x/(1 - x) - x^n;Transpose[Table[Rest[CoefficientList[Series[1/(1 - a), {x, 0, 10}], x]], {n, 1, 10}]] // Grid
%t (* Program translated from Maple: *)
%t T[n_, k_] := T[n, k] = If[n==0, 1, Sum[If[j==k, 0, T[n-j, k]], {j, 1, n}]];
%t Table[Table[T[n, d+1-n], {n, 1, d}], {d, 1, 12}] // Flatten (* _Jean-François Alcover_, Jun 06 2021, after _Alois P. Heinz_ *)
%K nonn,tabl
%O 1,8
%A _Geoffrey Critzer_, Jan 09 2011
%E Corrected and extended by _Alois P. Heinz_, Oct 23 2011
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