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Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=k*2^(k-1), read by columns.
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%I #33 Dec 17 2020 14:56:28

%S 1,1,2,2,1,5,12,18,20,18,15,11,6,3,1,15,64,166,332,566,864,1214,1596,

%T 1975,2320,2600,2780,2842,2780,2600,2320,1979,1608,1238,908,626,404,

%U 246,136,69,32,12,4,1,52,340,1315,3895,9770,21848,44880,86275,157140

%N Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=k*2^(k-1), read by columns.

%C T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.

%H Alois P. Heinz, <a href="/A326962/b326962.txt">Columns k = 0..10, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

%F Sum_{k=1..n} k * T(n,k) = A327115(n).

%F T(n*2^(n-1),n) = T(A001787(n),n) = 1.

%F T(n*2^(n-1)-1,n) = n for n >= 2.

%e T(4,3) = 12: 3abc1a, 3abc1b, 3abc1c, 2ab2ac, 2ab2bc, 2ac2bc, 2ab1a1c, 2ab1b1c, 2ac1a1b, 2ac1b1c, 2bc1a1b, 2bc1a1c.

%e Triangle T(n,k) begins:

%e 1;

%e 1;

%e 2;

%e 2, 5;

%e 1, 12, 15;

%e 18, 64, 52;

%e 20, 166, 340, 203;

%e 18, 332, 1315, 1866, 877;

%e 15, 566, 3895, 9930, 10710, 4140;

%e 11, 864, 9770, 39960, 74438, 64520, 21147;

%e 6, 1214, 21848, 134871, 386589, 564508, 408096, 115975;

%e ...

%p C:= binomial:

%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(

%p b(n-i*j, min(n-i*j, i-1), k)*C(C(k, i), j), j=0..n/i)))

%p end:

%p T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):

%p seq(seq(T(n, k), n=k..k*2^(k-1)), k=0..5);

%t c = Binomial;

%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i - 1], k] c[c[k, i], j], {j, 0, n/i}]]];

%t T[n_, k_] := Sum[b[n, n, i] (-1)^(k-i) c[k, i], {i, 0, k}];

%t Table[Table[T[n, k], {n, k, k 2^(k-1)}], {k, 0, 5}] // Flatten (* _Jean-François Alcover_, Dec 17 2020, after _Alois P. Heinz_ *)

%Y Main diagonal gives A000110.

%Y Row sums give A116539.

%Y Column sums give A003465.

%Y Cf. A001787, A255903, A326914 (this triangle read by rows), A327115, A327116, A327117.

%K nonn,tabf

%O 0,3

%A _Alois P. Heinz_, Sep 13 2019