cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A133710 Column l=3 of irregular triangle in A133709.

Original entry on oeis.org

0, 3, 35, 131, 347, 767, 1511, 2744, 4686, 7623, 11919, 18029, 26513, 38051, 53459, 73706, 99932, 133467, 175851, 228855, 294503, 375095, 473231, 591836, 734186, 903935, 1105143, 1342305, 1620381, 1944827
Offset: 1

Views

Author

N. J. A. Sloane, Dec 30 2007

Keywords

Crossrefs

Cf. A133709.

Programs

  • Maple
    A133710 := proc(m)
        A133709(m,3) ;
end proc:
seq(A133710(m),m=1..30) ; # R. J. Mathar, Nov 23 2011
  • Mathematica
    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, 3], {m, 1, 30}] (* Jean-François Alcover, Apr 03 2020 *)

Formula

Conjectures from Colin Barker, Apr 03 2020: (Start)
G.f.: x^2*(3 + 14*x - 51*x^2 + 60*x^3 - 31*x^4 + 6*x^5) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)

Extensions

Typo in data corrected by Jean-François Alcover, Apr 03 2020

A133711 Column l=4 of irregular triangle in A133709.

Original entry on oeis.org

0, 0, 140, 1435, 7693, 30450, 100330, 291265, 769015, 1883436, 4336320, 9475195, 19790605, 39733150, 77020830, 144681850, 264178990, 470096240, 817045500, 1389681375, 2317008105, 3792539410, 6102347050, 9663627675, 15077153821, 23197881100
Offset: 1

Views

Author

N. J. A. Sloane, Dec 30 2007

Keywords

Programs

  • Maple
    A133711 := proc(m)
        A133709(m,4) ;
end proc:
seq(A133711(m),m=1..30) ; # R. J. Mathar, Nov 23 2011
  • Mathematica
    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, 4], {m, 1, 30}] (* Jean-François Alcover, Apr 03 2020 *)

A133712 Column l=5 of irregular triangle in A133709.

Original entry on oeis.org

0, 0, 420, 15225, 185031, 1438906, 8689306, 44352346, 200070606, 818907792, 3093635652, 10914809127, 36278256537, 114357327402, 343708626298, 989318816383, 2737219679833, 7302776865288, 18839417766108, 47108352127209, 114421884019959
Offset: 1

Views

Author

N. J. A. Sloane, Dec 30 2007

Keywords

Crossrefs

Cf. A133709.

Programs

  • Maple
    A133712 := proc(m)
        A133709(m,5) ;
end proc:
seq(A133712(m),m=1..30) ; # R. J. Mathar, Nov 23 2011
  • Mathematica
    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, 5], {m, 1, 30}] (* Jean-François Alcover, Apr 03 2020 *)

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

Views

Author

N. J. A. Sloane, Dec 30 2007

Keywords

Examples

			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

Links

Crossrefs

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

Programs

  • Maple
    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
  • 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[, 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 *)

A006845 State assignments for n-state machine.

Original entry on oeis.org

0, 1, 3, 3, 140, 420, 840, 840, 10810800, 75675600, 454053600, 2270268000, 9081072000, 27243216000, 54486432000, 54486432000, 52401161274029568000, 786017419110443520000, 11004243867546209280000, 143055170278100720640000
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

After the initial 0, this sequence is formed by taking in turn the last 2^(n-1) elements of row n in the irregular triangle A133709. - Sean A. Irvine, Aug 14 2017

References

  • F. J. Hill and G. R. Peterson, Introduction to Switching Theory and Logical Design. Wiley, NY, 3rd ed., 1981, p. 308.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A007041, A133709.

Programs

  • Mathematica
    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}]];
Join[{{0}}, Table[Table[T[m, l], {l, 2^m - 2^(m-1), 2^m - 1}], {m, 1, 5}]] // Flatten (* Jean-François Alcover, Apr 03 2020 *)

Extensions

a(9) corrected and more terms from Sean A. Irvine, Aug 14 2017
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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