A253636 Second partial sums of eighth powers (A001016).

1, 258, 7076, 79430, 542409, 2685004, 10592400, 35277012, 103008345, 270739678, 652829892, 1464901802, 3092704433, 6196296120, 11862778432, 21824228040, 38761435089, 66718602714, 111659333380, 182200064046, 290563654073, 453803117636, 695353566480, 1046979329500

Offset

1,2

Comments

The general 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

G. C. Greubel, Table of n, a(n) for n = 1..1000

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 (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Formula

a(n) = (2*n^10 + 20*n^9 + 75*n^8 + 120*n^7 + 42*n^6 - 84*n^5 - 50*n^4 + 40*n^3 + 21*n^2 - 6*n)/180.

a(n) = 2*a(n-1) - a(n-2) + n^8. - Robert Israel, Jan 07 2015

G.f.: x*(1 + x)*(1 + 246*x + 4047*x^2 + 11572*x^3 + 4047*x^4 + 246*x^5 + x^6) / (1 - x)^11. - Bruno Berselli, Jan 08 2015

Maple

seq(n*(n+1)^2*(n+2)*(2*n^6 +12*n^5 +17*n^4 -12*n^3 -19*n^2 +18*n -3))/180, n=1..30); # G. C. Greubel, Aug 28 2019

Mathematica

Table[n(n+1)^2(n+2)(2n^6 +12n^5 +17n^4 -12n^3 -19n^2 +18n -3)/180, {n, 30}] (* Bruno Berselli, Jan 08 2015 *)
Nest[Accumulate, Range[30]^8, 2] (* or *) LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {1, 258, 7076, 79430, 542409, 2685004, 10592400, 35277012, 103008345, 270739678, 652829892}, 30] (* Harvey P. Dale, Jul 02 2017 *)

Prog

(Sage)
[(2*n^10+20*n^9+75*n^8+120*n^7+42*n^6-84*n^5-50*n^4+40*n^3+21*n^2-6*n)/180 for n in [1..24]] # Tom Edgar, Jan 07 2015
(Magma) [n*(n+1)^2*(n+2)*(2*n^6+12*n^5+17*n^4-12*n^3-19*n^2+18*n-3)/180: n in [1..25]]; // Bruno Berselli, Jan 08 2015
(PARI) a(n)=(2*n^10+20*n^9+75*n^8+120*n^7+42*n^6-84*n^5-50*n^4+40*n^3+21*n^2-6*n)/180 \\ Charles R Greathouse IV, Sep 08 2015
(GAP) List([1..30], n-> n*(n+1)^2*(n+2)*(2*n^6 +12*n^5 +17*n^4 -12*n^3 -19*n^2 +18*n -3))/180); # G. C. Greubel, Aug 28 2019

Crossrefs

Cf. A001016, A101089, A101092, A101093.

Sequence in context: A282060 A301546 A229330 * A271759 A228998 A219991

Adjacent sequences: A253633 A253634 A253635 * A253637 A253638 A253639

Keyword

nonn,easy

Author

Luciano Ancora, Jan 07 2015

Status

approved