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A326914 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), n>=0, min(j:A001787(j)>=n)<=k<=n, read by rows.

%I A326914 #50 Dec 17 2020 07:54:56
%S A326914 1,1,2,2,5,1,12,15,18,64,52,20,166,340,203,18,332,1315,1866,877,15,
%T A326914 566,3895,9930,10710,4140,11,864,9770,39960,74438,64520,21147,6,1214,
%U A326914 21848,134871,386589,564508,408096,115975,3,1596,44880,402756,1668338,3652712
%N A326914 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), n>=0, min(j:A001787(j)>=n)<=k<=n, read by rows.
%C A326914 T(n,k) is defined for all n>=0 and k>=0.  The triangle displays only positive terms.  All other terms are zero.
%H A326914 Alois P. Heinz, <a href="/A326914/b326914.txt">Rows n = 0..200, flattened</a>
%H A326914 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A326914 Sum_{k=1..n} k * T(n,k) = A327115(n).
%F A326914 T(n*2^(n-1),n) = T(A001787(n),n) = 1.
%F A326914 T(n*2^(n-1)-1,n) = n for n >= 2.
%e A326914 T(4,3) = 12: 3abc1a, 3abc1b, 3abc1c, 2ab2ac, 2ab2bc, 2ac2bc, 2ab1a1c, 2ab1b1c, 2ac1a1b, 2ac1b1c, 2bc1a1b, 2bc1a1c.
%e A326914 Triangle T(n,k) begins:
%e A326914   1;
%e A326914      1;
%e A326914         2;
%e A326914         2,  5;
%e A326914         1, 12,   15;
%e A326914            18,   64,    52;
%e A326914            20,  166,   340,    203;
%e A326914            18,  332,  1315,   1866,    877;
%e A326914            15,  566,  3895,   9930,  10710,   4140;
%e A326914            11,  864,  9770,  39960,  74438,  64520,  21147;
%e A326914             6, 1214, 21848, 134871, 386589, 564508, 408096, 115975;
%e A326914   ...
%p A326914 C:= binomial:
%p A326914 g:= proc(n) option remember; n*2^(n-1) end:
%p A326914 h:= proc(n) option remember; local k; for k from
%p A326914       `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od
%p A326914     end:
%p A326914 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A326914       b(n-i*j, min(n-i*j, i-1), k)*C(C(k, i), j), j=0..n/i)))
%p A326914     end:
%p A326914 T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):
%p A326914 seq(seq(T(n, k), k=h(n)..n), n=0..12);
%t A326914 c = Binomial;
%t A326914 g[n_] := g[n] = n*2^(n - 1);
%t A326914 h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0, h[n - 1]], True, k++, If[g[k] >= n, Return[k]]]];
%t A326914 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 A326914 T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}];
%t A326914 Table[Table[T[n, k], {k, h[n], n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 17 2020, after _Alois P. Heinz_ *)
%Y A326914 Main diagonal gives A000110.
%Y A326914 Row sums give A116539.
%Y A326914 Column sums give A003465.
%Y A326914 Cf. A001787, A255903, A326962 (this triangle read by columns), A327115, A327116, A327117.
%K A326914 nonn,tabf
%O A326914 0,3
%A A326914 _Alois P. Heinz_, Sep 13 2019