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 1-3 of 3 results.

A254643 Third partial sums of ninth powers (A001017).

Original entry on oeis.org

1, 515, 21225, 324275, 2862790, 17714466, 85232910, 339635850, 1168343775, 3571356685, 9906622271, 25333920885, 60457751900, 135939162100, 290221510860, 592024274916, 1159935330765, 2192313968775, 4011847886725, 7130537084615
Offset: 1

Views

Author

Luciano Ancora, Feb 05 2015

Keywords

Examples

			First differences:   1, 511, 19171, 242461, 1690981, ... (A022525)
------------------------------------------------------------------------
The ninth powers:    1, 512, 19683, 262144, 1953125, ... (A001017)
------------------------------------------------------------------------
First partial sums:  1, 513, 20196, 282340, 2235465, ... (A007487)
Second partial sums: 1, 514, 20710, 303050, 2538515, ... (A253637)
Third partial sums:  1, 515, 21225, 324275, 2862790, ... (this sequence)

Links

Crossrefs

Cf. A001017, A007487, A022525, A253637.

Programs

  • GAP
    List([1..30], n-> Binomial(n+3,4)*(2*n^8 +24*n^7 +98*n^6 +126*n^5 -97*n^4 -203*n^3 +127*n^2 +84*n -50)/110); # G. C. Greubel, Aug 28 2019
  • Magma
    [Binomial(n+3,4)*(2*n^8 +24*n^7 +98*n^6 +126*n^5 -97*n^4 -203*n^3 +127*n^2 +84*n -50)/110: n in [1..30]]; // G. C. Greubel, Aug 28 2019
  • Maple
    seq(binomial(n+3,4)*(2*n^8 +24*n^7 +98*n^6 +126*n^5 -97*n^4 -203*n^3 +127*n^2 +84*n -50)/110, n=1..30); # G. C. Greubel, Aug 28 2019
  • Mathematica
    Table[n(1+n)(2+n)(3+n)(-50 +84n +127n^2 -204n^3 -97n^4 +126n^5 +98n^6 +24n^7 +2n^8)/2640, {n, 20}] (* or *)
CoefficientList[Series[(1 +502x +14608x^2 +88234x^3 +156190x^4 +88234x^5 +14608x^6 +502x^7 +x^8)/(1-x)^13, {x, 0, 19}], x] (* Ancora *)
Accumulate[Accumulate[Accumulate[Range[10]^9]]] (* Alonso del Arte, Feb 09 2015 *)
  • PARI
    vector(30, n, m=n+3; binomial(m,4)*(2*(n*m)^4 -10*(n*m)^3 +11*(n*m)^2 +28*(n*m) -50)/110) \\ G. C. Greubel, Aug 28 2019
  • Sage
    [binomial(n+3,4)*(2*n^8 +24*n^7 +98*n^6 +126*n^5 -97*n^4 -203*n^3 +127*n^2 +84*n -50)/110 for n in (1..30)] # G. C. Greubel, Aug 28 2019

Formula

G.f.: x*(1 +502*x +14608*x^2 +88234*x^3 +156190*x^4 +88234*x^5 +14608*x^6 +502*x^7 +x^8)/(1-x)^13.
a(n) = n*(1+n)*(2+n)*(3+n)*(-50 +84*n +127*n^2 -204*n^3 -97*n^4 +126*n^5 +98*n^6 +24*n^7 +2*n^8)/2640.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + n^9.

Extensions

Edited by Alonso del Arte and Bruno Berselli, Feb 10 2015

A255179 Second differences of ninth powers (A001017).

Original entry on oeis.org

1, 510, 18660, 223290, 1448520, 6433590, 22151340, 63588210, 159338640, 359376750, 745368180, 1443884970, 2642886360, 4611828390, 7725765180, 12493804770, 19592282400, 29903014110, 44556993540, 64983894810, 92967744360, 130709124630, 180894272460
Offset: 0

Views

Author

Luciano Ancora, Feb 21 2015

Keywords

Examples

			Second differences:  1, 510, 18660, 223290, 1448520, ... (this sequence)
First differences:   1, 511, 19171, 242461, 1690981, ... (A022525)
------------------------------------------------------------------------
The ninth powers:    1, 512, 19683, 262144, 1953125, ... (A001017)
------------------------------------------------------------------------
First partial sums:  1, 513, 20196, 282340, 2235465, ... (A007487)
Second partial sums: 1, 514, 20710, 303050, 2538515, ... (A253637)
Third partial sums:  1, 515, 21225, 324275, 2862790, ... (A254643)

Links

Crossrefs

Cf. A001017, A007487, A022525, A253637, A254643, A255177, A255178.

Programs

  • Magma
    [1] cat [6*n*(3+28*n^2+42*n^4+12*n^6): n in [1..30]]; // Vincenzo Librandi, Mar 12 2015
  • Mathematica
    Join[{1}, Table[6 n (3 + 28 n^2 + 42 n^4 + 12 n^6), {n, 1, 30}]]
Join[{1},Differences[Range[0,30]^9,2]] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,510,18660,223290,1448520,6433590,22151340,63588210,159338640},30] (* Harvey P. Dale, Jan 26 2019 *)

Formula

G.f.: (1 + 502*x + 14608*x^2 + 88234*x^3 + 156190*x^4 + 88234*x^5 + 14608*x^6 + 502*x^7 + x^8)/(1 - x)^8.
a(n) = 6*n*(3 + 28*n^2 + 42*n^4 + 12*n^6) for n>0, a(0)=1.

Extensions

Corrected g.f. by Bruno Berselli, Feb 25 2015
Offset changed by Bruno Berselli, Mar 20 2015

A255183 Third differences of ninth powers (A001017).

Original entry on oeis.org

1, 509, 18150, 204630, 1225230, 4985070, 15717750, 41436870, 95750430, 200038110, 385991430, 698516790, 1199001390, 1968942030, 3113936790, 4768039590, 7098477630, 10310731710, 14653979430, 20426901270
Offset: 0

Views

Author

Luciano Ancora, Mar 18 2015

Keywords

Examples

			Third differences:   1, 509, 18150, 204630, 1225230, ...  (this sequence)
Second differences:  1, 510, 18660, 223290, 1448520, ...  (A255179)
First differences:   1, 511, 19171, 242461, 1690981, ...  (A022525)
---------------------------------------------------------------------
The ninth powers:    1, 512, 19683, 262144, 1953125, ...  (A001017)
---------------------------------------------------------------------

Links

Crossrefs

Cf. A001017, A007487, A022525, A253637, A255179, A255182.

Programs

  • Magma
    [1,509] cat [6*(84*n^6-252*n^5+630*n^4-840*n^3+756*n^2-378*n+85): n in [2..30]]; // Vincenzo Librandi, Mar 18 2015
  • Mathematica
    Join[{1, 509}, Table[6 (84 n^6 - 252 n^5 + 630 n^4 - 840 n^3 + 756 n^2 - 378 n + 85), {n, 2, 30}]]
Join[{1,509},Differences[Range[0,20]^9,3]] (* Harvey P. Dale, Apr 24 2015 *)

Formula

G.f.: (1 + 502*x + 14608*x^2 + 88234*x^3 + 156190*x^4 + 88234*x^5 + 14608*x^6 + 502*x^7 + x^8)/(1 - x)^7.
a(n) = 6*(84*n^6 - 252*n^5 + 630*n^4 - 840*n^3 + 756*n^2 - 378*n + 85) for n>1, a(0)=1, a(1)=509.

Extensions

Edited by Bruno Berselli, Mar 20 2015
Showing 1-3 of 3 results.

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