A210654 Triangle read by rows: T(n,k) (1 <= k <= n) = number of irreducible coverings by edges of the complete bipartite graph K_{n,k}.

1, 1, 2, 1, 6, 15, 1, 14, 48, 184, 1, 30, 165, 680, 2945, 1, 62, 558, 2664, 13080, 63756, 1, 126, 1827, 11032, 59605, 320292, 1748803, 1, 254, 5820, 46904, 281440, 1663248, 9791824, 58746304, 1, 510, 18177, 200232, 1379745, 8906544, 56499849, 361679040, 2361347073

Offset: 1

Author

N. J. A. Sloane, Mar 27 2012

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1,   2;
  1,   6,   15;
  1,  14,   48,   184;
  1,  30,  165,   680,   2945;
  1,  62,  558,  2664,  13080,   63756;
  1, 126, 1827, 11032,  59605,  320292, 1748803;
  1, 254, 5820, 46904, 281440, 1663248, 9791824, 58746304;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..120, flattened
• Ioan Tomescu, Some properties of irreducible coverings by cliques of complete multipartite graphs, J. Combin. Theory Ser. B 28 (1980), no. 2, 127--141. MR0572469 (81i:05106).

Crossrefs

Cf. A210655.

Programs

Maple

T:= proc(p, q) option remember; `if`(p=1 or q=1, 1,
         add(binomial(q, r)   *T(p-1, q-r), r=2..q-1)
      +q*add(binomial(p-1, s) *T(p-s-1, q-1), s=0..p-2))
    end:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Feb 10 2013

Mathematica

T[p_, q_] := T[p, q] = If[p == 1 || q == 1, 1, Sum[Binomial[q, r]*T[p-1, q-r], {r, 2, q-1}] + q*Sum[Binomial[p-1, s]*T[p-s-1, q-1], {s, 0, p-2}]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Mar 19 2014, after Alois P. Heinz *)

PARI

all(m) = {
mat = matrix(m, m);
for (i=1, m, for (j=1, m,
   if ((i == 1) || (j == 1), mat[i, j] = 1,
    if (i == j, mat[i, j] = i*mat[i-1,i-1] + sum(s=2,i-1, (s+1)*binomial(i,s)*mat[i-1,i-s]),
     mat[i, j] = sum(r=2, j-1, binomial(j,r)*mat[i-1,j-r]) + j*sum(s=0,i-2,binomial(i-1,s)*mat[i-s-1,j-1]));
   );
  );
);
for (i=1, m, for (j=1, i, print1(mat[i,j], ", ");); print(""););
print("");
for (i=1, m,print1(mat[i,i], ", "); );
} \\ Michel Marcus, Feb 10 2013

Formula

E.g.f.: exp(x*exp(y)+y*exp(x)-x-y-x*y)-1. - Alois P. Heinz, Feb 10 2013