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A242451 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.

%I A242451 #33 Oct 25 2018 14:44:26
%S A242451 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 A242451 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 A242451 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 A242451 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 A242451 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 A242451 T(n,n) = T(2n,n) = 1.
%C A242451 T(3n,n) = A244174(n).
%H A242451 Alois P. Heinz, <a href="/A242451/b242451.txt">Rows n = 0..140, flattened</a>
%e A242451 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 A242451 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 A242451 T(6,3) = 1: [2,2,2].
%e A242451 Triangle T(n,k) begins:
%e A242451   1;
%e A242451   0,   1;
%e A242451   0,   1,  1;
%e A242451   0,   3,  0,  1;
%e A242451   0,   6,  1,  0, 1;
%e A242451   0,  15,  0,  0, 0, 1;
%e A242451   0,  23,  7,  1, 0, 0, 1;
%e A242451   0,  53, 10,  0, 0, 0, 0, 1;
%e A242451   0,  94, 32,  0, 1, 0, 0, 0, 1;
%e A242451   0, 203, 31, 21, 0, 0, 0, 0, 0, 1;
%e A242451   0, 404, 71, 35, 0, 1, 0, 0, 0, 0, 1;
%p A242451 b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
%p A242451        b(n, i-1, p, k) +add(b(n-i*j, i-1, p+j, k)/j!,
%p A242451        j=max(1, k)..floor(n/i))))
%p A242451     end:
%p A242451 T:= (n, k)-> b(n$2, 0, k) -`if`(n=0 and k=0, 0, b(n$2, 0, k+1)):
%p A242451 seq(seq(T(n, k), k=0..n), n=0..14);
%t A242451 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 A242451 Columns k=0-10 give: A000007, A244164, A244165, A244166, A244167, A244168, A244169, A244170, A244171, A244172, A244173.
%Y A242451 Row sums give A011782.
%Y A242451 Cf. A242447 (the same for maximal multiplicity), A243978 (the same for partitions).
%K A242451 nonn,tabl,look
%O A242451 0,8
%A A242451 _Alois P. Heinz_, May 15 2014