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.

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A167565 A triangle related to the a(n) formulas for the rows of the ED2 array A167560.

Original entry on oeis.org

1, 2, 0, 3, 1, 2, 4, 4, 16, 0, 5, 10, 67, 14, 24, 6, 20, 202, 124, 368, 0, 7, 35, 497, 601, 2736, 444, 720, 8, 56, 1064, 2120, 13712, 6464, 16896, 0, 9, 84, 2058, 6096, 53121, 48876, 186732, 25584, 40320, 10, 120, 3684, 15168, 171258, 257640, 1350296
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 ED2 array A167560.
The recurrence relations for 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) = 2*n + 0.
Row 3: a(n) = 3*n^2 + 1*n + 2.
Row 4: a(n) = 4*n^3 + 4*n^2 + 16*n + 0.
Row 5: a(n) = 5*n^4 + 10*n^3 + 67*n^2 + 14*n + 24.
Row 6: a(n) = 6*n^5 + 20*n^4 + 202*n^3 + 124*n^2 + 368*n + 0.
Row 7: a(n) = 7*n^6 + 35*n^5 + 497*n^4 + 601*n^3 + 2736*n^2 + 444*n + 720.
Row 8: a(n) = 8*n^7 + 56*n^6 + 1064*n^5 + 2120*n^4 + 13712*n^3 + 6464*n^2 + 16896*n + 0.
Row 9: a(n) = 9*n^8 + 84*n^7 + 2058*n^6 + 6096*n^5 + 53121*n^4 + 48876*n^3 + 186732*n^2 + 25584*n + 40320.
Row 10: a(n) = 10*n^9 + 120*n^8 + 3684*n^7 + 15168*n^6 + 171258*n^5 + 257640*n^4 + 1350296*n^3 + 533472*n^2 + 1297152*n + 0.

Crossrefs

A167560 is the ED2 array.
A000012, A005843 (n=>1), 2*A104249 (n=>1), A167561, A167562 and A167563 equal the first sixth rows of the array.
A005359 equals the first right hand triangle column.
A000027, A000292, A167566, A167567 and A168304 equal the first five left hand triangle columns.
A000142 equals the row sums.
Cf. A003148 and A007318.

Extensions

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

A167567 The fourth left hand column of triangle A167565.

Original entry on oeis.org

0, 14, 124, 601, 2120, 6096, 15168, 33858, 69432, 132990, 240812, 415987, 690352, 1106768, 1721760, 2608548, 3860496, 5595006, 7957884, 11128205, 15323704, 20806720, 27890720, 36947430, 48414600, 62804430, 80712684
Offset: 4

Views

Author

Johannes W. Meijer, Nov 10 2009

Keywords

Links

Crossrefs

Equals the fourth left hand column of triangle A167565.
Other left hand columns are A000027, A000292, A167566 and A168304.

Programs

  • Mathematica
    LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 14, 124, 601,
  2120, 6096, 15168, 33858}, 100] (* G. C. Greubel, Jun 16 2016 *)

Formula

a(n) = (27*n^7 - 287*n^6 + 1113*n^5 - 1925*n^4 + 1428*n^3 - 308*n^2 - 48*n)/7!.
G.f.: (1*z^3 + 12*z^2 + 14*z + 0)/(1-z)^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) = 27.

Extensions

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

A168304 The fifth left hand column of triangle A167565.

Original entry on oeis.org

24, 368, 2736, 13712, 53121, 171258, 480711, 1210572, 2793219, 5996562, 12117677, 23257104, 42696758, 75408396, 128723898, 213203256, 343741122, 540958044, 832928118, 1257300704, 1863880095, 2717733590, 3902905305, 5526820260, 7725470805
Offset: 5

Views

Author

Johannes W. Meijer, Nov 23 2009

Keywords

Links

Crossrefs

Equals the fifth left hand column of triangle A167565.
Other left hand columns are A000027, A000292, A167566 and A167567.

Programs

  • Magma
    [(321*n^9-4500*n^8+25506*n^7-75096*n^6+121905*n^5- 104580*n^4+2736*n^2+37164*n^3-3456*n)/362880: n in [5..40]]; // Vincenzo Librandi, Jul 18 2016
  • Mathematica
    LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{24, 368, 2736, 13712, 53121, 171258, 480711, 1210572, 2793219, 5996562},50] (* G. C. Greubel, Jul 17 2016 *)

Formula

a(n) = (321*n^9 - 4500*n^8 + 25506*n^7 - 75096*n^6 + 121905*n^5 - 104580*n^4 + 2736*n^2 + 37164*n^3 - 3456*n)/9!.
G.f.: (z^4 + 32*z^3 + 136*z^2 + 128*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) = 321.
Showing 1-3 of 3 results.

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