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A253636 Second partial sums of eighth powers (A001016). 14
1, 258, 7076, 79430, 542409, 2685004, 10592400, 35277012, 103008345, 270739678, 652829892, 1464901802, 3092704433, 6196296120, 11862778432, 21824228040, 38761435089, 66718602714, 111659333380, 182200064046, 290563654073, 453803117636, 695353566480, 1046979329500 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..24.

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

MATHEMATICA

Table[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, 0, 25}] (* 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 [0..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

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

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Last modified November 16 07:06 EST 2018. Contains 317258 sequences. (Running on oeis4.)