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%I #43 Sep 08 2022 08:46:11
%S 1,260,7595,94360,723534,4037712,17944290,67127880,219319815,
%T 642251428,1718012933,4258676240,9892043980,21721707840,45414150132,
%U 90930820464,175208925885,326205634020,588861675535,1033717781096,1769137540730,2958360418000,4842936861750,7774492635000
%N Fourth partial sums of eighth powers (A001016).
%H Luciano Ancora, <a href="/A254647/b254647.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="/A254647/a254647_2.pdf"> Pascal’s triangle and recurrence relations for partial sums of m-th powers </a>
%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
%F G.f.: x*(1 +247*x +4293*x^2 +15619*x^3 +15619*x^4 +4293*x^5 +247*x^6 +x^7)/(1-x)^13.
%F a(n) = n*(1+n)*(2+n)^2*(3+n)*(4+n)*(1 +4*n +n^2)*(21 -48*n +20*n^2 + 16*n^3 +2*n^4)/23760.
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + n^8.
%e The eighth powers: 1, 256, 6561, 65536, 390625, ... (A001016)
%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, ... (A254642)
%e Fourth partial sums: 1, 260, 7595, 94360, 723534, ... (this sequence)
%p seq(binomial(n+4,5)*(n+2)*((n+2)^2-3)*(2*(n+2)^4 -28*(n+2)^2 +101)/198, n=1..30); # _G. C. Greubel_, Aug 28 2019
%t Table[n(1+n)(2+n)^2(3+n)(4+n)(1+4n+n^2)(21 -48n +20n^2 +16n^3 +2n^4 )/23760, {n,20}] (* or *)
%t Accumulate[Accumulate[Accumulate[Accumulate[Range[20]^8]]]] (* or *)
%t CoefficientList[Series[(1 +247x +4293x^2 +15619x^3 +15619x^4 +4293x^5 + 247x^6 +x^7)/(1-x)^13, {x,0,19}], x]
%o (PARI) a(n)=n*(1+n)*(2+n)^2*(3+n)*(4+n)*(1+4*n+n^2)*(21-48*n+20*n^2 +16*n^3+2*n^4)/23760 \\ _Charles R Greathouse IV_, Sep 08 2015
%o (PARI) vector(30, n, m=n+2; binomial(m+2,5)*m*(m^2-3)*(2*m^4-28*m^2 +101)/198)
%o (Magma) [Binomial(n+4,5)*(n+2)*((n+2)^2-3)*(2*(n+2)^4 -28*(n+2)^2 +101)/198: n in [1..30]]; // _G. C. Greubel_, Aug 28 2019
%o (Sage) [binomial(n+4,5)*(n+2)*((n+2)^2-3)*(2*(n+2)^4 -28*(n+2)^2 +101)/198 for n in (1..30)] # _G. C. Greubel_, Aug 28 2019
%o (GAP) List([1..30], n-> Binomial(n+4,5)*(n+2)*((n+2)^2-3)*(2*(n+2)^4 -28*(n+2)^2 +101)/198); # _G. C. Greubel_, Aug 28 2019
%Y Cf. A000542, A001016, A253636, A254642, A254644, A254645, A254646.
%K nonn,easy
%O 1,2
%A _Luciano Ancora_, Feb 05 2015
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