A133713 Array read by antidiagonals, giving the sizes pi_l(c_l(m,n)) of minimal covers (see reference for precise definition).
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, 1, 1, 28, 266, 995, 1371, 561, 50, 1, 1, 36, 462, 2541, 5901, 5087, 1277, 70, 1, 1, 45, 750, 5698, 20097, 30569, 17080, 2706, 95, 1
Offset: 2
Examples
Array begins: 1 1 1 1 1 1 1 1 1 ... 1 3 7 13 22 34 50 ... 1 6 25 81 226 561 1277 ... 1 10 65 325 1371 5087 17080 ... 1 15 140 995 5901 30569 142375 ... ...
Links
- A. P. Burger and J. H. van Vuuren, Balanced minimal covers of a finite set, Discr. Math. 307 (2007), 2853-2860.
Programs
-
Maple
A133713 := proc(l,cl) g := 1 ; for k from 1 to cl+1 do add( binomial(binomial(l,k+1)+i-1,i)*t^(i*k),i=0..ceil(cl/k)) ; g := g*% ; end do: g := expand(g) ; coeftayl(g,t=0,cl) ; end proc: seq(seq(A133713(d-k, k), k=0..d-2), d=2..11); # R. J. Mathar, Nov 23 2011
-
Mathematica
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[A133713%5Bl-cl+2,%20cl%5D,%20%7Bl,%200,%209%7D,%20%7Bcl,%200,%20l%7D%5D%20//%20Flatten%20(*%20_Jean-Fran%C3%A7ois%20Alcover">, 0] = 1; Table[A133713[l-cl+2, cl], {l, 0, 9}, {cl, 0, l}] // Flatten (* _Jean-François Alcover, Jan 07 2014, translated from Maple *)
Formula
Burger and van Vuuren give a generating function.
Extensions
Missing term 2706 inserted by Jean-François Alcover, Jan 07 2014