cp's OEIS Frontend

A327116 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 colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A327116 #28 Apr 27 2020 06:45:42
%S A327116 1,0,1,0,1,2,0,2,6,5,0,2,15,27,15,0,3,32,102,124,52,0,4,65,319,656,
%T A327116 600,203,0,5,124,897,2780,4210,3084,877,0,6,230,2346,10305,23040,
%U A327116 27567,16849,4140,0,8,414,5818,34864,108135,188284,186095,97640,21147
%N A327116 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 colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A327116 Alois P. Heinz, <a href="/A327116/b327116.txt">Rows n = 0..140, flattened</a>
%H A327116 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A327116 Sum_{k=1..n} k * T(n,k) = A327557(n).
%e A327116 T(3,2) = 6; 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.
%e A327116 Triangle T(n,k) begins:
%e A327116   1;
%e A327116   0, 1;
%e A327116   0, 1,   2;
%e A327116   0, 2,   6,    5;
%e A327116   0, 2,  15,   27,    15;
%e A327116   0, 3,  32,  102,   124,     52;
%e A327116   0, 4,  65,  319,   656,    600,    203;
%e A327116   0, 5, 124,  897,  2780,   4210,   3084,    877;
%e A327116   0, 6, 230, 2346, 10305,  23040,  27567,  16849,  4140;
%e A327116   0, 8, 414, 5818, 34864, 108135, 188284, 186095, 97640, 21147;
%e A327116   ...
%p A327116 C:= binomial:
%p A327116 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A327116       b(n-i*j, min(n-i*j, i-1), k)*C(C(k+i-1, i), j), j=0..n/i)))
%p A327116     end:
%p A327116 T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):
%p A327116 seq(seq(T(n, k), k=0..n), n=0..12);
%t A327116 c = Binomial;
%t A327116 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 - 1, i], j], {j, 0, n/i}]]];
%t A327116 T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}];
%t A327116 Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 27 2020, after _Alois P. Heinz_ *)
%Y A327116 Columns k=0-2 give: A000007, A000009 (for n>0), A327598.
%Y A327116 Main diagonal gives A000110.
%Y A327116 Row sums give A317776.
%Y A327116 T(2n,n) gives A327556.
%Y A327116 Cf. A255903, A326914, A326962, A327117, A327557, A309973.
%K A327116 nonn,tabl
%O A327116 0,6
%A A327116 _Alois P. Heinz_, Sep 13 2019