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
Luciano Ancora, Recurrence relation
Luciano Ancora, Second partial sums of m-th powers
FORMULA
a(n) = n*(n+1)^2*(n+2)*(n^2 + 2*n - 2)*(2*n^6 + 12*n^5 + 16*n^4 - 16*n^3 - 17*n^2 + 30*n - 5)/264.
a(n) = 2*a(n-1)-a(n-2)+n^10.
G.f.: x*(1 + 1013*x + 47840*x^2 + 455192*x^3 + 1310354*x^4 + 1310354*x^5 + 455192*x^6 + 47840*x^7 + 1013*x^8 + x^9)/(1-x)^13. - Vincenzo Librandi, Jan 19 2015
MATHEMATICA
a253710[n_] := Block[{f}, f[1] = 1; f[2] = 1026; f[x_] := 2*f[x - 1] - f[x - 2] + x^10; Array[f, n]]; a253710[21] (* Michael De Vlieger, Jan 11 2015 *)
CoefficientList[Series[(1 + 1013 x + 47840 x^2 + 455192 x^3 + 1310354 x^4 + 1310354 x^5 + 455192 x^6 + 47840 x^7 + 1013 x^8 + x^9) / (1 - x)^13, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 19 2015 *)
Nest[Accumulate, Range[30]^10, 2] (* Harvey P. Dale, May 10 2019 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Luciano Ancora, Jan 10 2015
STATUS
approved