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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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).

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: A123968 A282329 A343806 * A068797 A254639 A049951

Adjacent sequences:  A210651 A210652 A210653 * A210655 A210656 A210657

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Mar 27 2012

STATUS

approved

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Last modified December 7 20:19 EST 2021. Contains 349588 sequences. (Running on oeis4.)