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.

Previous Showing 11-13 of 13 results.

A253712 Second partial sums of 12th powers (A008456).

Original entry on oeis.org

1, 4098, 539636, 17852390, 279305769, 2717541484, 18997064400, 103996064052, 471424600185, 1838853136318, 6344710049172, 19766667410282, 56486709893873, 149900664752760, 373060957502272, 877696226962440, 1964953733652369, 4209042621768474, 8666446428950740, 17219850236133006, 33129081554701913, 61893315504320036
Offset: 1

Views

Author

Luciano Ancora, Jan 12 2015

Keywords

Comments

The formula for the second partial sums of m-th powers is: b(n,m) = (n+1)*F(m) - F(m+1), where F(m) are the m-th Faulhaber's formulas.

Links

Programs

  • Magma
    [(n+1)^2*n*(n+2)*(30*n^10+300*n^9+925*n^8+200*n^7-3022*n^6-772*n^5+7073*n^4-1228*n^3-7888*n^2+5528*n-691)/5460: n in [1..30]]; // Vincenzo Librandi, Jan 19 2015
  • Mathematica
    RecurrenceTable[{a[n] == 2 a[n - 1] - a[n - 2] + n^12, a[1] == 1, a[2] == 4098}, a, {n, 1, 25}] (* Bruno Berselli, Jan 19 2015 *)
Table[(n + 1)^2 n (n + 2) (30 n^10 + 300 n^9 + 925 n^8 + 200 n^7 - 3022 n^6 - 772 n^5 + 7073 n^4 - 1228 n^3 - 7888 n^2 + 5528 n - 691)/5460, {n, 1, 25}] (* Vincenzo Librandi, Jan 19 2015 *)
Nest[Accumulate[#]&,Range[30]^12,2] (* Harvey P. Dale, Aug 17 2020 *)

Formula

a(n) = (n+1)^2*n*(n+2)*(30*n^10+300*n^9+925*n^8+200*n^7-3022*n^6-772*n^5+7073*n^4-1228*n^3-7888*n^2+5528*n-691)/5460.
a(n) = 2*a(n-1)-a(n-2)+n^12.
G.f.: x*(1 + 4083*x + 478271*x^2 + 10187685*x^3 + 66318474*x^4 + 162512286*x^5 + 162512286*x^6 + 66318474*x^7 + 10187685*x^8 + 478271*x^9 + 4083*x^10 + x^11)/(1-x)^15. - Vincenzo Librandi, Jan 19 2015

Extensions

a(22) corrected by Vincenzo Librandi, Jan 19 2015

A253713 Second partial sums of 13th powers (A010801).

Original entry on oeis.org

1, 8194, 1610710, 70322090, 1359736595, 15709845116, 126948964044, 787943896860, 3990804658005, 17193665419150, 64919238324226, 219638016608374, 677231901484775, 1928540559615320, 5126044286105240, 12827147639965656, 30432829026732009, 68861475279169530, 149343104993864110, 311744734708558690, 628618742162372731
Offset: 1

Views

Author

Luciano Ancora, Jan 12 2015

Keywords

Comments

The formula for the second partial sums of m-th powers is: b(n,m) = (n+1)*F(m) - F(m+1), where F(m) are the m-th Faulhaber's formulas.

Links

Programs

  • Magma
    [n*(n+1)*(n+2)*(6*n^12+72*n^11+297*n^10+330*n^9-765*n^8-1368*n^7+2059*n^6+2994*n^5-4091*n^4-2724*n^3+4069*n^2+66*n-735)/1260: n in [1..30]]; // Vincenzo Librandi, Jan 19 2015
  • Mathematica
    Table[n (n+1) (n+2) (6 n^12 + 72 n^11 + 297 n^10 + 330 n^9 - 765 n^8 - 1368 n^7 + 2059 n^6 + 2994 n^5 - 4091 n^4 -2724 n^3 + 4069 n^2 + 66 n - 735) / 1260, {n, 40}] (* Vincenzo Librandi, Jan 19 2015 *)
Nest[Accumulate,Range[30]^13,2] (* Harvey P. Dale, Jul 24 2018 *)

Formula

a(n) = n*(n+1)*(n+2)*(6*n^12+72*n^11+297*n^10+330*n^9-765*n^8-1368*n^7+2059*n^6+2994*n^5-4091*n^4-2724*n^3+4069*n^2+66*n-735)/1260.
a(n) = 2*a(n-1)-a(n-2)+n^13.

A253711 Second partial sums of 11th powers (A008455).

Original entry on oeis.org

1, 2050, 181246, 4554746, 57756371, 473755052, 2867080476, 13850340492, 56214660117, 198578979742, 626254969978, 1796939330902, 4759784085863, 11772194010488, 27434359794488, 60688711622904, 128214959758953, 260009617974234, 508294535087734, 961379452201234, 1764741869856955, 3152422588924004, 5492913065904980
Offset: 1

Views

Author

Luciano Ancora, Jan 10 2015

Keywords

Comments

The formula for the second partial sums of m-th powers is: b(n,m) = (n+1)*F(m) - F(m+1), where F(m) are the m-th Faulhaber's formulas.

Links

Crossrefs

Cf. A008455.

Programs

  • Mathematica
    Table[n (n + 1) (n + 2) (70 n^10 + 700 n^9 + 2310 n^8 + 1680 n^7 - 4655 n^6 - 4410 n^5 + 8240 n^4 + 4120 n^3 - 7819 n^2 + 202 n + 1382)/10920, {n, 1, 20}] (* Vincenzo Librandi, Jan 15 2015 *)
RecurrenceTable[{a[n] == 2 a[n - 1] - a[n - 2] + n^11, a[1] == 1, a[2] == 2050}, a, {n, 1, 20}] (* Bruno Berselli, Jan 15 2015 *)

Formula

a(n) = n*(n+1)*(n+2)*(70*n^10 + 700*n^9 + 2310*n^8 + 1680*n^7 - 4655*n^6 - 4410*n^5 + 8240*n^4 + 4120*n^3 - 7819*n^2 + 202*n + 1382)/10920.
a(n) = 2*a(n-1) - a(n-2) + n^11.
G.f.: x*(1 + 2036*x + 152637*x^2 + 2203488*x^3 + 9738114*x^4 + 15724248*x^5 + 9738114*x^6 + 2203488*x^7 + 152637*x^8 + 2036*x^9 + x^10) / (1 - x)^14. - Vincenzo Librandi, Jan 15 2015
Previous Showing 11-13 of 13 results.

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