A133721 -id:A133721 - cp's OEIS frontend

A133722 Column 3 of triangle in A133721.

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 *)

A133724 Column 5 of triangle in A133721.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 22, 13, 7, 3, 1, 226, 81, 25, 6, 1, 1371, 325, 65, 10, 1, 5901, 995, 140, 15, 1, 20097, 2541, 266, 21, 1, 57813, 5698, 462, 28, 1, 146427, 11586, 750, 36, 1, 335742, 21825, 1155, 45, 1, 710677, 38665, 1705, 55, 1

Offset: 1

N. J. A. Sloane, Dec 30 2007

nonn

A133724 := proc(n)
        A133721(n,5) ;
end proc:
seq(A133724(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, 5], {m, 1, 60}] (* Jean-François Alcover, Apr 03 2020, after R. J. Mathar *)

A133733 Column 6 of triangle in A133721.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 34, 22, 13, 7, 3, 1, 561, 226, 81, 25, 6, 1, 5087, 1371, 325, 65, 10, 1, 30569, 5901, 995, 140, 15, 1, 138103, 20097, 2541, 266, 21, 1, 507360, 57813, 5698, 462, 28, 1, 1594272, 146427, 11586, 750, 36, 1, 4433484, 335742, 21825, 1155, 45, 1

Offset: 1

N. J. A. Sloane, Dec 30 2007

nonn

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, 6], {m, 1, 60}] (* Jean-François Alcover, Apr 03 2020, after R. J. Mathar *)

More terms from Jean-François Alcover, Apr 03 2020

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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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A133724 Column 5 of triangle in A133721.

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A133733 Column 6 of triangle in A133721.

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