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A326962 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.

%I A326962 #33 Dec 17 2020 14:56:28
%S A326962 1,1,2,2,1,5,12,18,20,18,15,11,6,3,1,15,64,166,332,566,864,1214,1596,
%T A326962 1975,2320,2600,2780,2842,2780,2600,2320,1979,1608,1238,908,626,404,
%U A326962 246,136,69,32,12,4,1,52,340,1315,3895,9770,21848,44880,86275,157140
%N A326962 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 A326962 T(n,k) is defined for all n>=0 and k>=0.  The triangle displays only positive terms.  All other terms are zero.
%H A326962 Alois P. Heinz, <a href="/A326962/b326962.txt">Columns k = 0..10, flattened</a>
%H A326962 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A326962 Sum_{k=1..n} k * T(n,k) = A327115(n).
%F A326962 T(n*2^(n-1),n) = T(A001787(n),n) = 1.
%F A326962 T(n*2^(n-1)-1,n) = n for n >= 2.
%e A326962 T(4,3) = 12: 3abc1a, 3abc1b, 3abc1c, 2ab2ac, 2ab2bc, 2ac2bc, 2ab1a1c, 2ab1b1c, 2ac1a1b, 2ac1b1c, 2bc1a1b, 2bc1a1c.
%e A326962 Triangle T(n,k) begins:
%e A326962   1;
%e A326962      1;
%e A326962         2;
%e A326962         2,  5;
%e A326962         1, 12,   15;
%e A326962            18,   64,    52;
%e A326962            20,  166,   340,    203;
%e A326962            18,  332,  1315,   1866,    877;
%e A326962            15,  566,  3895,   9930,  10710,   4140;
%e A326962            11,  864,  9770,  39960,  74438,  64520,  21147;
%e A326962             6, 1214, 21848, 134871, 386589, 564508, 408096, 115975;
%e A326962   ...
%p A326962 C:= binomial:
%p A326962 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A326962       b(n-i*j, min(n-i*j, i-1), k)*C(C(k, i), j), j=0..n/i)))
%p A326962     end:
%p A326962 T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):
%p A326962 seq(seq(T(n, k), n=k..k*2^(k-1)), k=0..5);
%t A326962 c = Binomial;
%t A326962 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 A326962 T[n_, k_] := Sum[b[n, n, i] (-1)^(k-i) c[k, i], {i, 0, k}];
%t A326962 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 A326962 Main diagonal gives A000110.
%Y A326962 Row sums give A116539.
%Y A326962 Column sums give A003465.
%Y A326962 Cf. A001787, A255903, A326914 (this triangle read by rows), A327115, A327116, A327117.
%K A326962 nonn,tabf
%O A326962 0,3
%A A326962 _Alois P. Heinz_, Sep 13 2019