A133710 -id:A133710 - cp's OEIS frontend

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

N. J. A. Sloane, Dec 30 2007

			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 ...
...

A. P. Burger and J. H. van Vuuren, Balanced minimal covers of a finite set, Discr. Math. 307 (2007), 2853-2860.

Rows give A002623, A133714-A133717.

Columns give A000217, A001296, A133718-A133710.

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

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 *)

Burger and van Vuuren give a generating function.

Missing term 2706 inserted by Jean-François Alcover, Jan 07 2014

A133709 Triangle read by rows: T(m,l) = number of labeled covers of size l of a finite set of m unlabeled elements (m >= 1, 1 <= l <= 2^m - 1).

Original entry on oeis.org

1, 1, 3, 3, 1, 7, 35, 140, 420, 840, 840, 1, 12, 131, 1435, 15225, 150570, 1351770, 10810800, 75675600, 454053600, 2270268000, 9081072000, 27243216000, 54486432000, 54486432000, 1, 18, 347, 7693, 185031, 4568046, 111793710, 2661422400

Offset: 1

N. J. A. Sloane, Dec 30 2007

			Triangle begins:
1
1 3 3
1 7 35 140 420 840 840
1 12 131 1435 15225 150570 1351770

A. P. Burger and J. H. van Vuuren, Balanced minimal covers of a finite set, Discr. Math. 307 (2007), 2853-2860.

Columns are given by A055998, A133710, A133711, A133712.

A133709 := proc(m,l)
        option remember;
        if l = 1 then
                1;
        else
                add((-1)^i*binomial(l,i)*binomial(2^(l-i)+m-2,m),i=0..l-1)
                - add(combinat[stirling2](l,i)*procname(m,i),i=1..l-1) ;
        end if;
end proc:
seq(seq(A133709(m,l),l=1..2^m-1),m=1..5) ; # R. J. Mathar, Nov 23 2011

T[m_, l_] := T[m, l] = If[l == 1, 1, Sum[(-1)^i Binomial[l, i] Binomial[ 2^(l-i)+m-2, m], {i, 0, l-1}] - Sum[StirlingS2[l, i] T[m, i], {i, 1, l-1} ] ];
Table[T[m, l], {m, 1, 5}, {l, 1, 2^m-1}] // Flatten (* Jean-François Alcover, Apr 01 2020, from Maple *)

Burger and van Vuuren give an explicit formula.

cp's OEIS Frontend

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

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