This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A255903 #34 Dec 17 2020 14:55:05
%S A255903 1,0,1,0,2,2,0,3,8,5,0,5,23,33,15,0,7,56,141,144,52,0,11,127,492,848,
%T A255903 675,203,0,15,268,1518,3936,5190,3396,877,0,22,547,4320,15800,30710,
%U A255903 32835,18270,4140,0,30,1072,11567,57420,154410,240012,216006,104656,21147
%N A255903 Number T(n,k) of collections of nonempty multisets with a total of n objects of exactly k colors; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A255903 T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.
%C A255903 In the case of exactly one color (k=1) each multiset of monochrome objects is fully described by its size and a collection of sizes corresponds to an integer partition. In the case of distinct colors for all objects (k=n) every multiset collection is a set partition.
%H A255903 Alois P. Heinz, <a href="/A255903/b255903.txt">Rows n = 0..140, flattened</a>
%F A255903 T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A075196(n,k-i).
%F A255903 Sum_{k=0..n} k * T(n,k) = A317178(n).
%e A255903 T(3,1) = 3: {{1},{1},{1}}, {{1},{1,1}}, {{1,1,1}}.
%e A255903 T(3,2) = 8: {{1},{1},{2}}, {{1},{2},{2}}, {{1},{1,2}}, {{1},{2,2}}, {{2},{1,1}}, {{2},{1,2}}, {{1,1,2}}, {{1,2,2}}.
%e A255903 T(3,3) = 5: {{1},{2},{3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1,2,3}}.
%e A255903 Triangle T(n,k) begins:
%e A255903 1;
%e A255903 0, 1;
%e A255903 0, 2, 2;
%e A255903 0, 3, 8, 5;
%e A255903 0, 5, 23, 33, 15;
%e A255903 0, 7, 56, 141, 144, 52;
%e A255903 0, 11, 127, 492, 848, 675, 203;
%e A255903 0, 15, 268, 1518, 3936, 5190, 3396, 877;
%e A255903 0, 22, 547, 4320, 15800, 30710, 32835, 18270, 4140;
%e A255903 ...
%p A255903 with(numtheory):
%p A255903 A:= proc(n, k) option remember; `if`(n=0, 1, add(A(n-j, k)*
%p A255903 add(d*binomial(d+k-1, k-1), d=divisors(j)), j=1..n)/n)
%p A255903 end:
%p A255903 T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p A255903 seq(seq(T(n, k), k=0..n), n=0..12);
%t A255903 A[n_, k_] := A[n, k] = If[n==0, 1, Sum[A[n-j, k]*Sum[d*Binomial[d+k-1, k-1], {d, Divisors[j]}], {j, 1, n}]/n]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i * Binomial[k, i], {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12} ] // Flatten (* _Jean-François Alcover_, Feb 20 2016, after _Alois P. Heinz_ *)
%Y A255903 Columns k=0-10 give: A000007, A000041 (for n>0), A255942, A255943, A255944, A255945, A255946, A255947, A255948, A255949, A255950.
%Y A255903 Main and lower diagonals give: A000110, A255951, A255952, A255953, A255954, A255955, A255956, A255957, A255958, A255959, A255960.
%Y A255903 Row sums give A255906.
%Y A255903 Antidiagonal sums give A258450.
%Y A255903 T(2n,n) gives A255907.
%Y A255903 Cf. A075196, A317178, A326914, A326962, A327116, A327117.
%K A255903 nonn,tabl
%O A255903 0,5
%A A255903 _Alois P. Heinz_, Mar 10 2015