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A133713 Array read by antidiagonals, giving the sizes pi_l(c_l(m,n)) of minimal covers (see reference for precise definition).

%I A133713 #10 Jan 07 2014 07:53:20
%S A133713 1,1,1,1,3,1,1,6,7,1,1,10,25,13,1,1,15,65,81,22,1,1,21,140,325,226,34,
%T A133713 1,1,28,266,995,1371,561,50,1,1,36,462,2541,5901,5087,1277,70,1,1,45,
%U A133713 750,5698,20097,30569,17080,2706,95,1
%N A133713 Array read by antidiagonals, giving the sizes pi_l(c_l(m,n)) of minimal covers (see reference for precise definition).
%H A133713 A. P. Burger and J. H. van Vuuren, <a href="http://dx.doi.org/10.1016/j.disc.2007.03.004">Balanced minimal covers of a finite set</a>, Discr. Math. 307 (2007), 2853-2860.
%F A133713 Burger and van Vuuren give a generating function.
%e A133713 Array begins:
%e A133713 1 1 1 1 1 1 1 1 1 ...
%e A133713 1 3 7 13 22 34 50 ...
%e A133713 1 6 25 81 226 561 1277 ...
%e A133713 1 10 65 325 1371 5087 17080 ...
%e A133713 1 15 140 995 5901 30569 142375 ...
%e A133713 ...
%p A133713 A133713 := proc(l,cl)
%p A133713         g := 1 ;
%p A133713         for k from 1 to cl+1 do
%p A133713           add( binomial(binomial(l,k+1)+i-1,i)*t^(i*k),i=0..ceil(cl/k)) ;
%p A133713           g := g*% ;
%p A133713         end do:
%p A133713         g := expand(g) ;
%p A133713         coeftayl(g,t=0,cl) ;
%p A133713 end proc:
%p A133713 seq(seq(A133713(d-k, k), k=0..d-2), d=2..11); # _R. J. Mathar_, Nov 23 2011
%t A133713 A133713[l_, cl_] := Module[{g, k, s}, g = 1; For[k = 1, k <= cl+1, k++, s = Sum[Binomial[Binomial[l, k+1]+i-1, i]*t^(i*k), {i, 0, Ceiling[cl/k]}]; g = g*s]; g = Expand[g]; SeriesCoefficient[g, {t, 0, cl}]]; A133713[_, 0] = 1; Table[A133713[l-cl+2, cl], {l, 0, 9}, {cl, 0, l}] // Flatten (* _Jean-François Alcover_, Jan 07 2014, translated from Maple *)
%Y A133713 Rows give A002623, A133714-A133717.
%Y A133713 Columns give A000217, A001296, A133718-A133710.
%K A133713 nonn,tabl
%O A133713 2,5
%A A133713 _N. J. A. Sloane_, Dec 30 2007
%E A133713 Missing term 2706 inserted by _Jean-François Alcover_, Jan 07 2014