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 1-4 of 4 results.

A167552 A triangle related to the a(n) formulas of the rows of the ED1 array A167546.

Original entry on oeis.org

1, 3, -2, 5, -5, 2, 7, -7, 14, -8, 9, -6, 63, -66, 24, 11, 0, 209, -264, 308, -144, 13, 13, 559, -689, 2236, -2132, 720, 15, 35, 1281, -1255, 11640, -14980, 14064, -5760, 17, 68, 2618, -1360, 47753, -68068, 145452, -126480, 40320
Offset: 1

Views

Author

Johannes W. Meijer, Nov 10 2009, Nov 23 2009

Keywords

Comments

The a(n) formulas given below correspond to the first ten rows of the ED1 array A167546.
The recurrence relations of the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A003148 and A007318.

Examples

			Row 1: a(n) = 1.
Row 2: a(n) = 3*n - 2.
Row 3: a(n) = 5*n^2 - 5*n + 2.
Row 4: a(n) = 7*n^3 - 7*n^2 + 14*n - 8.
Row 5: a(n) = 9*n^4 - 6*n^3 + 63*n^2 - 66*n + 24.
Row 6: a(n) = 11*n^5 + 0*n^4 + 209*n^3 - 264*n^2 + 308*n - 144.
Row 7: a(n) = 13*n^6 +13*n^5 +559*n^4 -689*n^3 +2236*n^2 -2132*n +720.
Row 8: a(n) = 15*n^7 + 35*n^6 + 1281*n^5 - 1255*n^4 + 11640*n^3 - 14980*n^2 + 14064*n - 5760.
Row 9: a(n) = 17*n^8 + 68*n^7 + 2618*n^6 - 1360*n^5 + 47753*n^4 - 68068*n^3 + 145452*n^2 - 126480*n + 40320.
Row 10: a(n) = 19*n^9 + 114*n^8 + 4902*n^7 + 684*n^6 + 163419*n^5 - 224694*n^4 + 1048268*n^3 - 1308264*n^2 + 1081632*n - 403200.

Crossrefs

A167546 is the ED1 array.
A000012, A016777, 2*A005891, A167547, A167548 and A167549 are the first sixth ED1 array rows.
A098557 and A167553 equal the first two right hand columns of this triangle.
A005408, A167554 and A167555, A168302 and A168303 equal the first five left hand columns of this triangle.
A000142 equals the row sums.
Cf. A003148 and A007318.

A167554 The second left hand column of triangle A167552.

Original entry on oeis.org

-2, -5, -7, -6, 0, 13, 35, 68, 114, 175, 253, 350, 468, 609, 775, 968, 1190, 1443, 1729, 2050, 2408, 2805, 3243, 3724, 4250, 4823, 5445, 6118, 6844, 7625, 8463, 9360, 10318, 11339, 12425, 13578, 14800, 16093, 17459, 18900, 20418
Offset: 2

Views

Author

Johannes W. Meijer, Nov 10 2009

Keywords

Links

Crossrefs

Equals the second left hand column of A167552.
Other left hand columns are A005408, A167555, A168302 and A168303.

Programs

  • Mathematica
    Table[(1/6)*(2*n^3 - 15*n^2 + 19*n - 6), {n,2,100}] (* or *) LinearRecurrence[{4,-6,4,-1}, {-2, -5, -7, -6}, 100] (* G. C. Greubel, Jun 15 2016 *)

Formula

a(n) = (2*n^3 - 15*n^2 + 19*n - 6)/3!.
G.f.: (z^2 + 3*z - 2)/(z-1)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) - 3*a(n-1) + 3*a(n-2) - a(n-3) = 2*1.

A168302 The fourth left hand column of triangle A167552.

Original entry on oeis.org

-8, -66, -264, -689, -1255, -1360, 684, 8502, 28842, 73150, 159588, 315549, 580723, 1010768, 1681640, 2694636, 4182204, 6314574, 9307264, 13429515, 19013709, 26465824, 36276980, 49036130, 65443950, 86327982, 112659084
Offset: 4

Views

Author

Johannes W. Meijer, Nov 23 2009

Keywords

Links

Crossrefs

Equals the fourth left hand column of triangle A167552.
Other left hand columns are A005408, A167554, A167555 and A168303.

Programs

  • Mathematica
    LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{-8, -66, -264, -689, -1255, -1360, 684, 8502},50] (* G. C. Greubel, Jul 17 2016 *)

Formula

a(n) = (54*n^7 - 1057*n^6 + 7245*n^5 - 24535*n^4 + 45801*n^3 - 47488*n^2 + 25020*n - 5040)/7!.
G.f.: (z^4 + 23*z^3 + 40*z^2 - 2*z - 8)/(z-1)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
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) = 2*27.

A168303 The fifth left hand column of triangle A167552.

Original entry on oeis.org

24, 308, 2236, 11640, 47753, 163419, 485121, 1284987, 3101175, 6927921, 14502059, 28718989, 54217878, 98183330, 171418854, 289756194, 475873962, 761609034, 1190854830, 1823151902, 2738088199, 4040638965, 5867589455, 8395197525, 11848267665
Offset: 5

Views

Author

Johannes W. Meijer, Nov 23 2009

Keywords

Links

Crossrefs

Equals the fifth left hand column of triangle A167552.
Other left hand columns are A005408, A167554, A167555 and A168302.

Programs

  • Magma
    [(642*n^9-13833*n^8+132840*n^7-726642*n^6+ 2439738*n^5-5133177*n^4+6699660*n^3-5194188*n^2+ 2157840*n- 362880)/362880: n in [5..40]]; // Vincenzo Librandi, Jul 18 2016
  • Mathematica
    LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{24, 308, 2236, 11640, 47753, 163419, 485121, 1284987, 3101175, 6927921},50] (* G. C. Greubel, Jul 17 2016 *)

Formula

a(n) = (642*n^9 - 13833*n^8 + 132840*n^7 - 726642*n^6 + 2439738*n^5 - 5133177*n^4 + 6699660*n^3 - 5194188*n^2 + 2157840*n - 362880)/9!
G.f.: (z^5 + 53*z^4 + 260*z^3 + 236*z^2 + 68*z + 24)/(1-z)^10.
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10).
a(n) - 9*a(n-1) + 36*a(n-2) - 84*a(n-3) + 126*a(n-4) - 126*a(n-5) + 84*a(n-6) - 36*a(n-7) + 9*a(n-8) - a(n-9) = 2*321.
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.