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

Author

Luciano Ancora, Jan 10 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

• Luciano Ancora, Recurrence relation
• Luciano Ancora, Second partial sums of m-th powers
• Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.

Programs

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 *)

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