A291774 Triangle read by rows: chromatic invariant T(n,m) of the complete bipartite graph K_{m,n}.

1, 0, 1, 0, 1, 5, 0, 1, 13, 73, 0, 1, 29, 301, 2069, 0, 1, 61, 1081, 11581, 95401, 0, 1, 125, 3613, 57749, 673261, 6487445, 0, 1, 253, 11593, 268381, 4306681, 55213453, 610093513, 0, 1, 509, 36301, 1191989, 25794781, 431525429, 6077248381, 75796724309, 0, 1, 1021, 111961, 5136061, 147587401, 3173843821, 56153444761, 864806272861, 12020754177001

Offset

1,6

Links

Table of n, a(n) for n=1..55.

Eric Weisstein's World of Mathematics, Chromatic Invariant

Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Formula

T(m,n) = Sum_{k = 0..m-1} k!*(-1)^(k + m)*(k + 1)^n*Stirling2(m, k + 2) for max(m,n) > 1.

Example

Triangle begins:
1
0 1
0 1 5
0 1 13 73
0 1 29 301 2069

Mathematica

Join[{1}, Table[Sum[k! (-1)^(k + m) (k + 1)^n StirlingS2[m, k + 2], {k, 0, m - 1}], {n, 2, 10}, {m, n}]] // Flatten

Crossrefs

Main diagonal gives A048144.

Sequence in context: A221308 A241855 A221800 * A222061 A345453 A064315

Adjacent sequences: A291771 A291772 A291773 * A291775 A291776 A291777

Keyword

nonn,tabl

Author

Eric W. Weisstein, Aug 31 2017

Status

approved