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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}.
2
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
1,3
LINKS
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).
FORMULA
E.g.f.: exp(x*exp(y)+y*exp(x)-x-y-x*y)-1. - Alois P. Heinz, Feb 10 2013
EXAMPLE
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;
...
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 *)
PROG
(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
CROSSREFS
Cf. A210655.
Sequence in context: A282329 A343806 A372254 * A068797 A254639 A049951
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Mar 27 2012
STATUS
approved