cp's OEIS Frontend

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}.

%I A210654 #34 May 27 2019 12:11:06
%S A210654 1,1,2,1,6,15,1,14,48,184,1,30,165,680,2945,1,62,558,2664,13080,63756,
%T A210654 1,126,1827,11032,59605,320292,1748803,1,254,5820,46904,281440,
%U A210654 1663248,9791824,58746304,1,510,18177,200232,1379745,8906544,56499849,361679040,2361347073
%N 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}.
%H A210654 Alois P. Heinz, <a href="/A210654/b210654.txt">Rows n = 1..120, flattened</a>
%H A210654 Ioan Tomescu, <a href="https://doi.org/10.1016/0095-8956(80)90060-X">Some properties of irreducible coverings by cliques of complete multipartite graphs</a>, J. Combin. Theory Ser. B 28 (1980), no. 2, 127--141. MR0572469 (81i:05106).
%F A210654 E.g.f.: exp(x*exp(y)+y*exp(x)-x-y-x*y)-1. - _Alois P. Heinz_, Feb 10 2013
%e A210654 Triangle begins:
%e A210654   1;
%e A210654   1,   2;
%e A210654   1,   6,   15;
%e A210654   1,  14,   48,   184;
%e A210654   1,  30,  165,   680,   2945;
%e A210654   1,  62,  558,  2664,  13080,   63756;
%e A210654   1, 126, 1827, 11032,  59605,  320292, 1748803;
%e A210654   1, 254, 5820, 46904, 281440, 1663248, 9791824, 58746304;
%e A210654   ...
%p A210654 T:= proc(p, q) option remember; `if`(p=1 or q=1, 1,
%p A210654          add(binomial(q, r)   *T(p-1, q-r), r=2..q-1)
%p A210654       +q*add(binomial(p-1, s) *T(p-s-1, q-1), s=0..p-2))
%p A210654     end:
%p A210654 seq(seq(T(n, k), k=1..n), n=1..12);  # _Alois P. Heinz_, Feb 10 2013
%t A210654 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_ *)
%o A210654 (PARI) all(m) = {
%o A210654 mat = matrix(m, m);
%o A210654 for (i=1, m, for (j=1, m,
%o A210654    if ((i == 1) || (j == 1), mat[i, j] = 1,
%o A210654     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]),
%o A210654      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]));
%o A210654    );
%o A210654   );
%o A210654 );
%o A210654 for (i=1, m, for (j=1, i, print1(mat[i,j], ", ");); print(""););
%o A210654 print("");
%o A210654 for (i=1, m,print1(mat[i,i], ", "); );
%o A210654 } \\ _Michel Marcus_, Feb 10 2013
%Y A210654 Cf. A210655.
%K A210654 nonn,tabl
%O A210654 1,3
%A A210654 _N. J. A. Sloane_, Mar 27 2012