cp's OEIS Frontend

A384968 Triangle read by rows: T(n,k) is the number of proper vertex colorings of the n-complete bipartite graph using exactly k interchangeable colors, 2 <= k <= 2*n.

%I A384968 #6 Jun 18 2025 23:17:21
%S A384968 1,1,2,1,1,6,11,6,1,1,14,61,86,50,12,1,1,30,275,770,927,530,150,20,1,
%T A384968 1,62,1141,5710,12160,12632,6987,2130,355,30,1,1,126,4571,38626,
%U A384968 134981,228382,209428,110768,34902,6580,721,42,1,1,254,18061,248766,1367310,3553564,4989621,4093126,2061782,655788,132958,16996,1316,56,1
%N A384968 Triangle read by rows: T(n,k) is the number of proper vertex colorings of the n-complete bipartite graph using exactly k interchangeable colors, 2 <= k <= 2*n.
%C A384968 Permuting the colors does not change the coloring. T(n,k) is the number of ways to partition the vertices into k independent sets.
%H A384968 Andrew Howroyd, <a href="/A384968/b384968.txt">Table of n, a(n) for n = 1..2500</a> (rows 1..50)
%H A384968 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>.
%H A384968 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VertexColoring.html">Vertex Coloring</a>.
%F A384968 T(n,k) = Sum_{j=1..k-1} Stirling2(n,j)*Stirling2(n,k-j).
%F A384968 T(n,k) = A274310(2*n-1, k-1).
%e A384968 Triangle begins (n >= 1, k >= 2):
%e A384968   1;
%e A384968   1,  2,    1;
%e A384968   1,  6,   11,    6,     1;
%e A384968   1, 14,   61,   86,    50,    12,    1;
%e A384968   1, 30,  275,  770,   927,   530,  150,   20,   1;
%e A384968   1, 62, 1141, 5710, 12160, 12632, 6987, 2130, 355, 30, 1;
%e A384968   ...
%o A384968 (PARI) T(n,k) = sum(j=1, k-1, stirling(n,j,2)*stirling(n,k-j,2))
%o A384968 for(n=1, 7, print(vector(2*n-1,k,T(n,k+1))))
%Y A384968 Row sums are A001247.
%Y A384968 Columns k=2..5 are A000012, A000918, A384980, A384981.
%Y A384968 Cf. A008277, A212084, A274310.
%K A384968 nonn,tabf
%O A384968 1,3
%A A384968 _Andrew Howroyd_, Jun 18 2025