login
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.
14
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, 0, 42, 74, 21, 1, 0, 56, 116, 37, 2, 0, 77, 175, 67, 5, 0, 101, 263, 112, 10, 0, 135, 385, 187, 20, 0, 176, 560, 302, 40, 1, 0, 231, 800, 479, 72, 2, 0, 297, 1135, 741, 127, 5
OFFSET
0,5
COMMENTS
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).
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
LINKS
FORMULA
T(n,k) = 1/k! * Sum_{i=0..k} (-1)^i*binomial(k,i) A321884(n,k-i).
T(n*(n+1)/2,n) = T(A000217(n),n) = 1.
T(n*(n+3)/2,n) = T(A000096(n),n) = A000712(n).
Sum_{k=1..A003056(n)} k * T(n,k) = A322304(n).
EXAMPLE
T(6,1) = 11: 111111a, 2a1111a, 22a11a, 222a, 3a111a, 3a2a1a, 33a, 4a11a, 4a2a, 5a1a, 6a.
T(6,2) = 9: 2a1111b, 22a11b, 3a111b, 3a2a1b, 3a2b1a, 3a2b1b, 4a11b, 4a2b, 5a1b.
T(6,3) = 1: 3a2b1c.
Triangle T(n,k) begins:
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;
0, 42, 74, 21, 1;
0, 56, 116, 37, 2;
0, 77, 175, 67, 5;
...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
(t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!:
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..20);
MATHEMATICA
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[n_, k_] := Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]/k!;
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 *)
CROSSREFS
Columns k=0-10 give: A000007, A000041 (for n>0), A327285, A327286, A327287, A327288, A327289, A327290, A327291, A327292, A327293.
Row sums give A305106.
Sequence in context: A144955 A225624 A168020 * A225084 A238345 A299070
KEYWORD
nonn,look,tabf
AUTHOR
Alois P. Heinz, Aug 27 2019
STATUS
approved