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A327117 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that a color pattern for part i has i distinct colors in increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A327117 #29 Oct 04 2019 13:10:27
%S A327117 1,0,1,0,1,2,0,1,4,5,0,1,7,18,15,0,1,10,45,84,52,0,1,14,94,298,415,
%T A327117 203,0,1,18,174,844,1995,2178,877,0,1,23,300,2081,7440,13638,12131,
%U A327117 4140,0,1,28,486,4652,23670,64898,95823,71536,21147,0,1,34,756,9682,67390,259599,566447,694676,445356,115975
%N A327117 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that a color pattern for part i has i distinct colors in increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A327117 The sequence of column k satisfies a linear recurrence with constant coefficients of order k*2^(k-1) = A001787(k).
%H A327117 Alois P. Heinz, <a href="/A327117/b327117.txt">Rows n = 0..140, flattened</a>
%H A327117 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A327117 Sum_{k=1..n} k * T(n,k) = A327118(n).
%e A327117 T(3,2) = 4: 2ab1a, 2ab1b, 1a1a1b, 1a1b1b.
%e A327117 Triangle T(n,k) begins:
%e A327117   1;
%e A327117   0, 1;
%e A327117   0, 1,  2;
%e A327117   0, 1,  4,   5;
%e A327117   0, 1,  7,  18,   15;
%e A327117   0, 1, 10,  45,   84,    52;
%e A327117   0, 1, 14,  94,  298,   415,    203;
%e A327117   0, 1, 18, 174,  844,  1995,   2178,    877;
%e A327117   0, 1, 23, 300, 2081,  7440,  13638,  12131,   4140;
%e A327117   0, 1, 28, 486, 4652, 23670,  64898,  95823,  71536,  21147;
%e A327117   0, 1, 34, 756, 9682, 67390, 259599, 566447, 694676, 445356, 115975;
%e A327117   ...
%p A327117 C:= binomial:
%p A327117 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A327117       b(n-i*j, min(n-i*j, i-1), k)*C(C(k, i)+j-1, j), j=0..n/i)))
%p A327117     end:
%p A327117 T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):
%p A327117 seq(seq(T(n, k), k=0..n), n=0..12);
%t A327117 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] Binomial[Binomial[k, i] + j - 1, j], {j, 0, n/i}]]];
%t A327117 T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
%t A327117 Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 04 2019, after _Alois P. Heinz_ *)
%Y A327117 Columns k=0-3 give: A000007, A057427, A014616(n-1) for n>1, A327842.
%Y A327117 Main diagonal gives A000110.
%Y A327117 Row sums give A116540.
%Y A327117 T(2n,n) gives A327843.
%Y A327117 Cf. A001787, A255903, A326914, A326962, A327116, A327118.
%K A327117 nonn,tabl
%O A327117 0,6
%A A327117 _Alois P. Heinz_, Sep 13 2019