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.

A167591 A triangle related to the a(n) formulas of the rows of the ED4 array A167584.

Original entry on oeis.org

1, 4, -2, 12, -8, 9, 32, -16, 120, -60, 80, 0, 952, -768, 525, 192, 160, 5664, -5008, 12396, -5670, 448, 896, 27888, -20672, 162740, -133128, 72765, 1024, 3584, 120064, -46720, 1537216, -1562464, 2557296, -1081080, 2304, 12288, 467712, 76800
Offset: 1

Views

Author

Johannes W. Meijer, Nov 10 2009

Keywords

Comments

The a(n) formulas given below correspond to the first ten rows of the ED4 array A167584.
The recurrence relations of the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A013609, A003148, A081277 and A079628.

Examples

			Row 1: a(n) = 1.
Row 2: a(n) = 4*n - 2.
Row 3: a(n) = 12*n^2 - 8*n + 9.
Row 4: a(n) = 32*n^3 - 16*n^2 + 120*n - 60.
Row 5: a(n) = 80*n^4 + 0*n^3 + 952*n^2 - 768*n + 525.
Row 6: a(n) = 192*n^5 + 160*n^4 + 5664*n^3 - 5008*n^2 + 12396*n - 5670.
Row 7: a(n) = 448*n^6 + 896*n^5 + 27888*n^4 - 20672*n^3 + 162740*n^2 - 133128*n + 72765.
Row 8: a(n) = 1024*n^7 + 3584*n^6 + 120064*n^5 - 46720*n^4 + 1537216*n^3 - 1562464*n^2 + 2557296*n - 1081080.
Row 9: a(n) = 2304*n^8 + 12288*n^7 + 467712*n^6 + 76800*n^5 + 11589216*n^4 - 12058368*n^3 + 47963568*n^2 - 38278080*n + 18243225.
Row 10: a(n) = 5120*n^9 + 38400*n^8 + 1686528*n^7 + 1540608*n^6 + 73898880*n^5 - 66179520*n^4 + 631348672*n^3 - 669559008*n^2 + 869709780*n - 344594250.

Crossrefs

A167584 is the ED4 array.
A000012, A016825, A167585, A167586 and A167587 equal the first five rows of the ED4 array.
A001787, A167592, A167593, A168307 and A168308 equal the first five left hand triangle columns.
A001193 equals the first right hand triangle column.
A024199 equals the row sums.
Cf. A013609, A003148, A079628 and A081277.

Extensions

Comment and formulas added by Johannes W. Meijer, Nov 23 2009

A167592 The second left hand column of triangle A167591.

Original entry on oeis.org

-2, -8, -16, 0, 160, 896, 3584, 12288, 38400, 112640, 315392, 851968, 2236416, 5734400, 14417920, 35651584, 86900736, 209190912, 498073600, 1174405120, 2745171968, 6366953472, 14663286784, 33554432000, 76336332800, 172738215936
Offset: 2

Views

Author

Johannes W. Meijer, Nov 10 2009

Keywords

Links

Crossrefs

Equals the second left hand column of triangle A167591.
Other left hand columns are A001787, A167593, A168307 and A168308.

Programs

  • Mathematica
    LinearRecurrence[{8, -24, 32, -16}, {-2, -8, -16, 0}, 100] (* G. C. Greubel, Jun 17 2016 *)

Formula

a(n) = 2^n*(n^3 - 6*n^2 + 5*n)/12.
GF(z) = (8*z - 2)/(1-2*z)^4.
a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4).
a(n) - 7*a(n-1) + 18*a(n-2) - 20*a(n-3) + 8*a(n-4) = 1*2^(n-2).

Extensions

Formulae and links added by Johannes W. Meijer, Nov 23 2009

A168307 The fourth left hand column of triangle A167591.

Original entry on oeis.org

-60, -768, -5008, -20672, -46720, 76800, 1540608, 10610688, 55114752, 246005760, 992808960, 3720331264, 13156941824, 44395134976, 144054681600, 452151214080, 1379061202944, 4102054477824, 11934819680256, 34047283691520, 95430020956160, 263252302888960
Offset: 4

Views

Author

Johannes W. Meijer, Nov 23 2009, Nov 25 2009

Keywords

Links

Crossrefs

Equals the fourth left hand column of triangle A167591.
Other left hand columns are A001787, A167592, A167593 and A168308.

Programs

  • Magma
    [2^n*(27*n^7-434*n^6+2289*n^5-5705*n^4+7938*n^3- 6461*n^2+2346*n)/10080: n in [4..30]]; // Vincenzo Librandi, Jul 18 2016
  • Mathematica
    LinearRecurrence[{16,-112,448,-1120,1792,-1792,1024,-256}, {-60, -768, -5008, -20672, -46720, 76800, 1540608, 10610688}, 50] (* G. C. Greubel, Jul 17 2016 *)

Formula

a(n) = 2^n*(27*n^7 - 434*n^6 + 2289*n^5 - 5705*n^4 + 7938*n^3 - 6461*n^2 + 2346*n)/10080.
G.f.: (320*z^3 + 560*z^2 + 192*z - 60)/(1-2*z)^8.
a(n) = 16*a(n-1) - 112*a(n-2) + 448*a(n-3) - 1120*a(n-4) + 1792*a(n-5) - 1792*a(n-6) + 1024*a(n-7) - 256*a(n-8).
a(n) - 15*a(n-1) + 98*a(n-2) - 364*a(n-3) + 840*a(n-4) - 1232*a(n-5) + 1120*a(n-6) - 576*a(n-7) + 128*a(n-8) = 27*2^(n-2).

A168308 The fifth left hand column of triangle A167591.

Original entry on oeis.org

525, 12396, 162740, 1537216, 11589216, 73898880, 413745024, 2087500800, 9672309504, 41745859584, 169680276480, 655126331392, 2419298385920, 8593269522432, 29494166618112, 98195558891520, 318148898783232, 1005877391523840, 3110695891894272
Offset: 5

Views

Author

Johannes W. Meijer, Nov 23 2009

Keywords

Links

Crossrefs

Equals the fifth left hand column of triangle A167591.
Other left hand columns are A001787, A167592, A167593 and A168307.

Programs

  • Magma
    [2^n*(107*n^9-1824*n^8+14124*n^7-62538*n^6+ 165228*n^5-259476*n^4+241561*n^3-133542*n^2+ 36360*n)/241920: n in [5..30]]; // Vincenzo Librandi, Jul 18 2016
  • Mathematica
    LinearRecurrence[{20,-180,960,-3360,8064,-13440,15360,-11520,5120,-1024},{525, 12396, 162740, 1537216, 11589216, 73898880, 413745024, 2087500800, 9672309504, 41745859584}, 50] (* G. C. Greubel, Jul 17 2016 *)

Formula

a(n) = 2^n*(107*n^9 - 1824*n^8 + 14124*n^7 - 62538*n^6 + 165228*n^5 - 259476*n^4 + 241561*n^3 - 133542*n^2 + 36360*n)/241920.
G.f.: (1936*z^4 + 9696*z^3 + 9320*z^2 + 1896*z + 525)/(1-2*z)^10.
a(n) = 20*a(n-1) - 180*a(n-2) + 960*a(n-3) - 3360*a(n-4) + 8064*a(n-5) - 13440*a(n-6) + 15360*a(n-7) - 11520*a(n-8) + 5120*a(n-9) - 1024*a(n-10).
a(n) - 19*a(n-1) + 162*a(n-2) - 816*a(n-3) + 2688*a(n-4) - 6048*a(n-5) + 9408*a(n-6) -9984*a(n-7) + 6912*a(n-8) - 2816*a(n-9) + 512*a(n-10) = 321*2^(n-2).
Showing 1-4 of 4 results.

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