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

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

Offset: 1

Author

Luciano Ancora, Jan 12 2015

Keywords

• nonn
• easy

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

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Luciano Ancora, Recurrence relation for the second partial sums of m-th powers
• Luciano Ancora, Second partial sums of the m-th powers

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.