A253711 Second partial sums of 11th powers (A008455).

1, 2050, 181246, 4554746, 57756371, 473755052, 2867080476, 13850340492, 56214660117, 198578979742, 626254969978, 1796939330902, 4759784085863, 11772194010488, 27434359794488, 60688711622904, 128214959758953, 260009617974234, 508294535087734, 961379452201234, 1764741869856955, 3152422588924004, 5492913065904980

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..23.

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

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

Index entries for linear recurrences with constant coefficients, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Formula

a(n) = n*(n+1)*(n+2)*(70*n^10 + 700*n^9 + 2310*n^8 + 1680*n^7 - 4655*n^6 - 4410*n^5 + 8240*n^4 + 4120*n^3 - 7819*n^2 + 202*n + 1382)/10920.

a(n) = 2*a(n-1) - a(n-2) + n^11.

G.f.: x*(1 + 2036*x + 152637*x^2 + 2203488*x^3 + 9738114*x^4 + 15724248*x^5 + 9738114*x^6 + 2203488*x^7 + 152637*x^8 + 2036*x^9 + x^10) / (1 - x)^14. - Vincenzo Librandi, Jan 15 2015

Mathematica

Table[n (n + 1) (n + 2) (70 n^10 + 700 n^9 + 2310 n^8 + 1680 n^7 - 4655 n^6 - 4410 n^5 + 8240 n^4 + 4120 n^3 - 7819 n^2 + 202 n + 1382)/10920, {n, 1, 20}] (* Vincenzo Librandi, Jan 15 2015 *)
RecurrenceTable[{a[n] == 2 a[n - 1] - a[n - 2] + n^11, a[1] == 1, a[2] == 2050}, a, {n, 1, 20}] (* Bruno Berselli, Jan 15 2015 *)

Crossrefs

Cf. A008455.

Sequence in context: A045059 A224119 A183842 * A168176 A224728 A224721

Adjacent sequences: A253708 A253709 A253710 * A253712 A253713 A253714

Keyword

nonn,easy

Author

Luciano Ancora, Jan 10 2015

Status

approved