A253710 Second partial sums of tenth powers (A008454).

1, 1026, 61100, 1169750, 12044025, 83384476, 437200176, 1864757700, 6779099625, 21693441550, 62545208076, 165314338826, 405941961425, 935824239000, 2042356907200, 4248401203176, 8470439399601, 16262944822650, 30186516503500, 54350088184350, 95193540843401, 162596916293876, 271426802958000, 443660070587500

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..24.

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

Sequence in context: A221008 A282254 A229332 * A271761 A229000 A168153

Adjacent sequences: A253707 A253708 A253709 * A253711 A253712 A253713

Keyword

nonn,easy

Author

Luciano Ancora, Jan 10 2015

Status

approved