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Third partial sums of eighth powers (A001016).
6

%I #47 Oct 14 2015 04:18:14

%S 1,259,7335,86765,629174,3314178,13906578,49183590,152191935,

%T 422931613,1075761505,2540663307,5633367740,11829663860,23692442292,

%U 45516670332,84278105421,150996708135,262656041515,444856105561,735419759634,1189222877270

%N Third partial sums of eighth powers (A001016).

%H Luciano Ancora, <a href="/A254642/b254642.txt">Table of n, a(n) for n = 1..1000</a>

%H Luciano Ancora, <a href="/A254640/a254640_1.pdf">Partial sums of m-th powers with Faulhaber polynomials</a>

%H Luciano Ancora, <a href="/A254642/a254642_2.pdf"> Pascal’s triangle and recurrence relations for partial sums of m-th powers </a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

%F G.f.: (x + 247*x^2 + 4293*x^3 + 15619*x^4 + 15619*x^5 + 4293*x^6 + 247*x^7 + x^8)/(- 1 + x)^12.

%F a(n) = n*(1 + n)*(2 + n)*(3 + n)*(3 + 2*n)*(1 + 36*n - 69*n^2 + 45*n^4 + 18*n^5 + 2*n^6)/3960.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + n^8.

%e First differences: 1, 255, 6305, 58975, 325089, ...(A022524)

%e --------------------------------------------------------------------

%e The eighth powers: 1, 256, 6561, 65536, 390625, ...(A001016)

%e --------------------------------------------------------------------

%e First partial sums: 1, 257, 6818, 72354, 462979, ...(A000542)

%e Second partial sums: 1, 258, 7076, 79430, 542409, ...(A253636)

%e Third partial sums: 1, 259, 7335, 86765, 629174, ...(this sequence)

%t Table[n (1 + n) (2 + n) (3 + n) (3 + 2 n) (1 + 36 n - 69 n^2 + 45 n^4 + 18 n^5 + 2 n^6)/3960, {n, 22}]

%t Accumulate[Accumulate[Accumulate[Range[22]^8]]]

%t CoefficientList[Series[(1 + 247 x + 4293 x^2 + 15619 x^3 + 15619 x^4 + 4293 x^5 + 247 x^6 + x^7)/(- 1 + x)^12, {x, 0, 22}], x]

%o (PARI) a(n)=n*(1+n)*(2+n)*(3+n)*(3+2*n)*(1+36*n-69*n^2+45*n^4+18*n^5+2*n^6)/3960 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A000542, A001016, A022524, A253636.

%K nonn,easy

%O 1,2

%A _Luciano Ancora_, Feb 05 2015