A133713 -id:A133713 - cp's OEIS frontend

A133714 Row l=4 of array in A133713.

1, 6, 25, 81, 226, 561, 1277, 2706, 5411, 10297, 18788, 33043, 56277, 93148, 150303, 237029, 366135, 554993, 826904, 1212695, 1752761, 2499432, 3519935, 4899801, 6747052, 9196983, 12417949, 16617932, 22052383, 29033069, 37938516

Offset: 0

N. J. A. Sloane, Dec 30 2007

nonn

A133714 := proc(c)
        A133713(4,c) ;
end proc:
seq(A133714(c),c=0..30) ; # R. J. Mathar, Nov 23 2011

A133717 Row l=7 of array in A133713.

Original entry on oeis.org

1, 21, 266, 2541, 20097, 138103, 849514, 4771889, 24828755, 120944516, 556107055, 2429461391, 10137839293, 40585353434, 156451707633, 582555425669, 2100914443649, 7355423560191, 25050878785976, 83145222843357, 269368091719619

Offset: 0

N. J. A. Sloane, Dec 30 2007

nonn

A133717 := proc(c)
        A133713(7,c) ;
end proc:
seq(A133717(c),c=0..30) ; # R. J. Mathar, Nov 23 2011

A133718 Column 3 of array in A133713.

Original entry on oeis.org

1, 13, 81, 325, 995, 2541, 5698, 11586, 21825, 38665, 65131, 105183, 163891, 247625, 364260, 523396, 736593, 1017621, 1382725, 1850905, 2444211, 3188053, 4111526, 5247750, 6634225, 8313201, 10332063, 12743731, 15607075

Offset: 2

N. J. A. Sloane, Dec 30 2007

nonn

A133718 := proc(l)
        A133713(l,3) ;
end proc:
seq(A133718(l),l=2..30) ; # R. J. Mathar, Nov 23 2011

A133715 Row l=5 of array in A133713.

Original entry on oeis.org

1, 10, 65, 325, 1371, 5087, 17080, 52835, 152606, 415602, 1075410, 2660010, 6320476, 14485642, 32131580, 69181744, 144943566, 296136752, 591142760, 1154839554, 2211164332, 4154912452, 7671082770, 13930509344, 24906380837, 43880045952

Offset: 0

N. J. A. Sloane, Dec 30 2007

nonn

A133715 := proc(c)
        A133713(5,c) ;
end proc:
seq(A133715(c),c=0..30) ; # R. J. Mathar, Nov 23 2011

A133716 Row l=6 of array in A133713.

Original entry on oeis.org

1, 15, 140, 995, 5901, 30569, 142375, 607700, 2410081, 8972559, 31608089, 106027825, 340416406, 1050541069, 3127340719, 9007681389, 25168555576, 68374424019, 180958790169, 467385754264, 1179927541924, 2915558734984, 7060185950979

Offset: 0

N. J. A. Sloane, Dec 30 2007

nonn

A133716 := proc(c)
        A133713(6,c) ;
end proc:
seq(A133716(c),c=0..30) ; # R. J. Mathar, Nov 23 2011

A133719 Column 4 of array in A133713.

Original entry on oeis.org

1, 22, 226, 1371, 5901, 20097, 57813, 146427, 335742, 710677, 1408693, 2643004, 4732728, 8141238, 13524078, 21787914, 34162095, 52284504, 78303484, 114997729, 165916135, 235539711, 329467755, 454630605, 619531380, 834519231

Offset: 2

N. J. A. Sloane, Dec 30 2007

nonn

A133719 := proc(l)
        A133713(l,4) ;
end proc:
seq(A133719(l),l=2..30) ; # R. J. Mathar, Nov 23 2011

A133720 Column 5 of array in A133713.

Original entry on oeis.org

1, 34, 561, 5087, 30569, 138103, 507360, 1594272, 4433484, 11176033, 25989535, 56493801, 115960208, 226569266, 424098592, 764503872, 1332959307, 2256042444, 3717882133, 5981238571, 9414651939, 14526981953, 22010865680, 32796846160

Offset: 2

N. J. A. Sloane, Dec 30 2007

nonn

A133720 := proc(l)
        A133713(l,5) ;
end proc:
seq(A133720(l),l=2..30) ; # R. J. Mathar, Nov 23 2011

A133721 Triangle read by rows: T(m,n) = number of n-balanced and minimal labeled covers of a finite set of m unlabeled elements (m >= 1, 1 <= n <= m).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 7, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 10, 1, 13, 1, 1, 1, 1, 1, 1, 1, 25, 7, 1, 1, 1, 1, 1, 1, 1, 15, 6, 3, 22, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 21, 65, 81, 7, 34, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10

Offset: 1

N. J. A. Sloane, Dec 30 2007

			Triangle begins:
1
1 1
1 1 1
1 1 1 1
1 3 1 1 1
1 1 1 1 1 1
1 6 7 1 1 1 1
1 1 3 1 1 1 1 1
1 10 1 13 1 1 1 1 1
1 1 25 7 1 1 1 1 1 1
1 15 6 3 22 1 1 1 1 1 1

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

Cf. A133709. Column n=2 is essentially A000217. Columns 3, 4, 5, 6 give A133722, A133723, A133724, A133733.

A133721 := proc(m,n)
        l := ceil(m/n) ;
        c := n*ceil(m/n)-m ;
        A133713(l,c) ;
end proc: # 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[, 0] = 1; a[m, n_] := A133713[Ceiling[m/n], n*Ceiling[m/n] - m]; Table[a[m, n], {m, 1, 14}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jan 20 2014, after R. J. Mathar *)

A133722 Column 3 of triangle in A133721.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 7, 3, 1, 25, 6, 1, 65, 10, 1, 140, 15, 1, 266, 21, 1, 462, 28, 1, 750, 36, 1, 1155, 45, 1, 1705, 55, 1, 2431, 66, 1, 3367, 78, 1, 4550, 91, 1, 6020, 105, 1, 7820, 120, 1, 9996, 136, 1, 12597, 153, 1, 15675

Offset: 1

N. J. A. Sloane, Dec 30 2007

nonn

A133722 := proc(n)
        A133721(n,3) ;
end proc:
seq(A133722(n),n=1..60) ; # 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]; SeriesCoefficient[g, {t, 0, cl}]];
a[m_, n_] := A133713[Ceiling[m/n], n*Ceiling[m/n] - m];
Table[a[m, 3], {m, 1, 55}] (* Jean-François Alcover, Apr 03 2020, after R. J. Mathar *)

Conjectures from Colin Barker, Apr 03 2020: (Start)
G.f.: x^3*(1 + x + x^2 - 4*x^3 + 2*x^4 - 2*x^5 + 6*x^6 + x^8 - 4*x^9 + x^12) / ((1 - x)^5*(1 + x + x^2)^5).
a(n) = 5*a(n-3) - 10*a(n-6) + 10*a(n-9) - 5*a(n-12) + a(n-15) for n>14.
(End)

A133723 Column 4 of triangle in A133721.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 13, 7, 3, 1, 81, 25, 6, 1, 325, 65, 10, 1, 995, 140, 15, 1, 2541, 266, 21, 1, 5698, 462, 28, 1, 11586, 750, 36, 1, 21825, 1155, 45, 1, 38665, 1705, 55, 1, 65131, 2431, 66, 1, 105183, 3367, 78, 1, 163891, 4550, 91, 1

Offset: 1

N. J. A. Sloane, Dec 30 2007

nonn

A133723 := proc(n)
        A133721(n,4) ;
end proc:
seq(A133723(n),n=1..60) ; # 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]; SeriesCoefficient[g, {t, 0, cl}]];
a[m_, n_] := A133713[Ceiling[m/n], n*Ceiling[m/n] - m];
Table[a[m, 4], {m, 1, 60}] (* Jean-François Alcover, Apr 03 2020, after R. J. Mathar *)

cp's OEIS Frontend

A133714 Row l=4 of array in A133713.

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A133717 Row l=7 of array in A133713.

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A133718 Column 3 of array in A133713.

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A133715 Row l=5 of array in A133713.

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A133716 Row l=6 of array in A133713.

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A133719 Column 4 of array in A133713.

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A133720 Column 5 of array in A133713.

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A133721 Triangle read by rows: T(m,n) = number of n-balanced and minimal labeled covers of a finite set of m unlabeled elements (m >= 1, 1 <= n <= m).

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A133722 Column 3 of triangle in A133721.

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A133723 Column 4 of triangle in A133721.

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