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%I #33 Oct 25 2018 14:44:26
%S 1,0,1,0,1,1,0,3,0,1,0,6,1,0,1,0,15,0,0,0,1,0,23,7,1,0,0,1,0,53,10,0,
%T 0,0,0,1,0,94,32,0,1,0,0,0,1,0,203,31,21,0,0,0,0,0,1,0,404,71,35,0,1,
%U 0,0,0,0,1,0,855,77,91,0,0,0,0,0,0,0,1,0,1648,222,105,71,0,1,0,0,0,0,0,1
%N Number T(n,k) of compositions of n in which the minimal multiplicity of parts equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C T(0,0) = 1 by convention. T(n,k) counts the compositions of n in which at least one part has multiplicity k and no part has a multiplicity smaller than k.
%C T(n,n) = T(2n,n) = 1.
%C T(3n,n) = A244174(n).
%H Alois P. Heinz, <a href="/A242451/b242451.txt">Rows n = 0..140, flattened</a>
%e T(5,1) = 15: [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1], [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], [3,1,1], [2,3], [3,2], [1,4], [4,1], [5].
%e T(6,2) = 7: [1,1,2,2], [1,2,1,2], [1,2,2,1], [2,1,1,2], [2,1,2,1], [2,2,1,1], [3,3].
%e T(6,3) = 1: [2,2,2].
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 3, 0, 1;
%e 0, 6, 1, 0, 1;
%e 0, 15, 0, 0, 0, 1;
%e 0, 23, 7, 1, 0, 0, 1;
%e 0, 53, 10, 0, 0, 0, 0, 1;
%e 0, 94, 32, 0, 1, 0, 0, 0, 1;
%e 0, 203, 31, 21, 0, 0, 0, 0, 0, 1;
%e 0, 404, 71, 35, 0, 1, 0, 0, 0, 0, 1;
%p b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
%p b(n, i-1, p, k) +add(b(n-i*j, i-1, p+j, k)/j!,
%p j=max(1, k)..floor(n/i))))
%p end:
%p T:= (n, k)-> b(n$2, 0, k) -`if`(n=0 and k=0, 0, b(n$2, 0, k+1)):
%p seq(seq(T(n, k), k=0..n), n=0..14);
%t b[n_, i_, p_, k_] := b[n, i, p, k] = If[n == 0, p!, If[i < 1, 0, b[n, i - 1, p, k] + Sum[b[n - i*j, i - 1, p + j, k]/j!, {j, Max[1, k], Floor[n/i]}]]]; T[n_, k_] := b[n, n, 0, k] - If[n == 0 && k == 0, 0, b[n, n, 0, k + 1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Jan 27 2015, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000007, A244164, A244165, A244166, A244167, A244168, A244169, A244170, A244171, A244172, A244173.
%Y Row sums give A011782.
%Y Cf. A242447 (the same for maximal multiplicity), A243978 (the same for partitions).
%K nonn,tabl,look
%O 0,8
%A _Alois P. Heinz_, May 15 2014
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