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A321878 Number T(n,k) of partitions of n into colored blocks of equal parts, such that all colors from a set of size k are used and the colors are introduced in increasing order; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

%I A321878 #66 Dec 14 2020 07:49:22
%S A321878 1,0,1,0,2,0,3,1,0,5,2,0,7,5,0,11,9,1,0,15,17,2,0,22,28,5,0,30,47,10,
%T A321878 0,42,74,21,1,0,56,116,37,2,0,77,175,67,5,0,101,263,112,10,0,135,385,
%U A321878 187,20,0,176,560,302,40,1,0,231,800,479,72,2,0,297,1135,741,127,5
%N A321878 Number T(n,k) of partitions of n into colored blocks of equal parts, such that all colors from a set of size k are used and the colors are introduced in increasing order; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.
%C A321878 T(n,k) is defined for all n>=0 and k>=0.  The triangle contains only elements with 0 <= k <= A003056(n).  T(n,k) = 0 for k > A003056(n).
%C A321878 For fixed k>=1, T(n,k) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2, 1-k))*n/3)) * sqrt(Pi^2 - 6*polylog(2, 1-k)) / (4*k!*sqrt(3*k)*Pi*n). - _Vaclav Kotesovec_, Sep 18 2019
%H A321878 Alois P. Heinz, <a href="/A321878/b321878.txt">Rows n = 0..1000, flattened</a>
%F A321878 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^i*binomial(k,i) A321884(n,k-i).
%F A321878 T(n*(n+1)/2,n) = T(A000217(n),n) = 1.
%F A321878 T(n*(n+3)/2,n) = T(A000096(n),n) = A000712(n).
%F A321878 Sum_{k=1..A003056(n)} k * T(n,k) = A322304(n).
%e A321878 T(6,1) = 11: 111111a, 2a1111a, 22a11a, 222a, 3a111a, 3a2a1a, 33a, 4a11a, 4a2a, 5a1a, 6a.
%e A321878 T(6,2) = 9: 2a1111b, 22a11b, 3a111b, 3a2a1b, 3a2b1a, 3a2b1b, 4a11b, 4a2b, 5a1b.
%e A321878 T(6,3) = 1: 3a2b1c.
%e A321878 Triangle T(n,k) begins:
%e A321878   1;
%e A321878   0,  1;
%e A321878   0,  2;
%e A321878   0,  3,   1;
%e A321878   0,  5,   2;
%e A321878   0,  7,   5;
%e A321878   0, 11,   9,  1;
%e A321878   0, 15,  17,  2;
%e A321878   0, 22,  28,  5;
%e A321878   0, 30,  47, 10;
%e A321878   0, 42,  74, 21, 1;
%e A321878   0, 56, 116, 37, 2;
%e A321878   0, 77, 175, 67, 5;
%e A321878   ...
%p A321878 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A321878      (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
%p A321878     end:
%p A321878 T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!:
%p A321878 seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..20);
%t A321878 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
%t A321878 T[n_, k_] := Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]/k!;
%t A321878 Table[Table[T[n, k], {k, 0, Floor[(Sqrt[1 + 8n] - 1)/2]}], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%Y A321878 Columns k=0-10 give: A000007, A000041 (for n>0), A327285, A327286, A327287, A327288, A327289, A327290, A327291, A327292, A327293.
%Y A321878 Row sums give A305106.
%Y A321878 Cf. A000096, A000217, A000712, A003056, A321884, A322304.
%K A321878 nonn,look,tabf
%O A321878 0,5
%A A321878 _Alois P. Heinz_, Aug 27 2019