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

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

Offset

1,2

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

Table of n, a(n) for n=1..22.

Luciano Ancora, Recurrence relation for the second partial sums of m-th powers

Luciano Ancora, Second partial sums of the m-th powers

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

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 *)

Prog

(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

Crossrefs

Sequence in context: A345507 A346357 A282548 * A345605 A346279 A023354

Adjacent sequences: A253709 A253710 A253711 * A253713 A253714 A253715

Keyword

nonn

Author

Luciano Ancora, Jan 12 2015

Extensions

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

Status

approved