A253710 - OEIS

%I #59 May 10 2019 15:15:17

%S 1,1026,61100,1169750,12044025,83384476,437200176,1864757700,

%T 6779099625,21693441550,62545208076,165314338826,405941961425,

%U 935824239000,2042356907200,4248401203176,8470439399601,16262944822650,30186516503500,54350088184350,95193540843401,162596916293876,271426802958000,443660070587500

%N Second partial sums of tenth powers (A008454).

%C 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.

%H Luciano Ancora, <a href="/A253636/a253636_3.pdf">Recurrence relation</a>

%H Luciano Ancora, <a href="/A253636/a253636_4.pdf">Second partial sums of m-th powers</a>

%F 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.

%F a(n) = 2*a(n-1)-a(n-2)+n^10.

%F 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

%t 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 *)

%t 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 *)

%t Nest[Accumulate,Range[30]^10,2] (* _Harvey P. Dale_, May 10 2019 *)

%K nonn,easy

%O 1,2

%A _Luciano Ancora_, Jan 10 2015