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 A256894 #30 Mar 07 2025 06:39:33
%S A256894 1,1,1,1,2,1,1,4,4,1,1,8,13,7,1,1,16,40,35,11,1,1,32,121,155,80,16,1,
%T A256894 1,64,364,651,490,161,22,1,1,128,1093,2667,2751,1316,294,29,1,1,256,
%U A256894 3280,10795,14721,9597,3108,498,37,1,1,512,9841,43435,76630,65352
%N A256894 Triangle read by rows, T(n,k) = Sum_{j=0..n-k+1} C(n-1,j-1)*T(n-j,k-1) if k != 0 else 1, n>=0, 0<=k<=n.
%C A256894 Can be understood as a convolution matrix or as a sequence-to-triangle transformation similar to the partial Bell polynomials defined as: S -> T(n, k, S) = Sum_{j=0..n-k+1} C(n-1,j-1)*S(j)*T(n-j,k-1,S) if k != 0 else S(0)^n. Here S(n) = 1 for all n. The case S(n) = n gives the triangle of idempotent numbers A059297.
%C A256894 From _Manfred Boergens_, Mar 04 2025: (Start)
%C A256894 T(n,k) = number of collections of k+1 disjoint [n]-subsets covering [n], with [0]={}.
%C A256894 For disjoint covers (collections without an empty set) see A008277.
%C A256894 For non-disjoint collections see A163353.
%C A256894 For non-disjoint covers see A055154. (End)
%F A256894 From _Manfred Boergens_, Mar 04 2025: (Start)
%F A256894 T(n,k) = S2(n,k) + S2(n,k+1).
%F A256894 T(n,k) = A008277(n,k) + A008277(n,k+1) for n>=1, k>=1. (End)
%e A256894 Triangle starts:
%e A256894 1;
%e A256894 1, 1;
%e A256894 1, 2, 1;
%e A256894 1, 4, 4, 1;
%e A256894 1, 8, 13, 7, 1;
%e A256894 1, 16, 40, 35, 11, 1;
%e A256894 1, 32, 121, 155, 80, 16, 1;
%e A256894 1, 64, 364, 651, 490, 161, 22, 1;
%e A256894 The signed version is the inverse of A326326:
%e A256894 1;
%e A256894 -1, 1;
%e A256894 1, -2, 1;
%e A256894 -1, 4, -4, 1;
%e A256894 1, -8, 13, -7, 1;
%e A256894 -1, 16, -40, 35, -11, 1;
%e A256894 1, -32, 121, -155, 80, -16, 1;
%e A256894 -1, 64, -364, 651, -490, 161, -22, 1. - _Peter Luschny_, Jul 02 2019
%e A256894 T(4,3)=7 is the number of disjoint [4]-covering collections of 4 subsets:
%e A256894 {{1},{2},{3},{4}}
%e A256894 {{1,2},{3},{4},{}}
%e A256894 {{1,3},{2},{4},{}}
%e A256894 {{1,4},{2},{3},{}}
%e A256894 {{2,3},{1},{4},{}}
%e A256894 {{2,4},{1},{3},{}}
%e A256894 {{3,4},{1},{2},{}}. - _Manfred Boergens_, Mar 04 2025
%p A256894 # Implemented as a sequence transformation acting on f: n -> 1,1,1,1,... .
%p A256894 F := proc(n, k, f) option remember; `if`(k=0, f(0)^n,
%p A256894 add(binomial(n-1,j-1)*f(j)*F(n-j,k-1,f),j=0..n-k+1)) end:
%p A256894 for n from 0 to 7 do seq(F(n,k,j->1), k=0..n) od;
%t A256894 Table[StirlingS2[n, m+1]+StirlingS2[n, m], {n, 0, 10}, {m, 0, n}]//Flatten (* _Manfred Boergens_, Mar 04 2025 *)
%Y A256894 Row sums are A186021.
%Y A256894 Partial row sums are A381682.
%Y A256894 T(n+1,1) = A000079(n).
%Y A256894 T(n+1,n) = A000124(n).
%Y A256894 Cf. A059297, A256895, A326326.
%Y A256894 Cf. A008277, A055154, A163353.
%K A256894 nonn,tabl,easy
%O A256894 0,5
%A A256894 _Peter Luschny_, Apr 28 2015