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 A254647 Fourth partial sums of eighth powers (A001016). 15
 1, 260, 7595, 94360, 723534, 4037712, 17944290, 67127880, 219319815, 642251428, 1718012933, 4258676240, 9892043980, 21721707840, 45414150132, 90930820464, 175208925885, 326205634020, 588861675535, 1033717781096, 1769137540730, 2958360418000, 4842936861750, 7774492635000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Luciano Ancora, Table of n, a(n) for n = 1..1000 Luciano Ancora, Partial sums of m-th powers with Faulhaber polynomials Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1). FORMULA 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. 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. a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + n^8. EXAMPLE The eighth powers:   1, 256, 6561, 65536, 390625, ... (A001016) First partial sums:  1, 257, 6818, 72354, 462979, ... (A000542) Second partial sums: 1, 258, 7076, 79430, 542409, ... (A253636) Third partial sums:  1, 259, 7335, 86765, 629174, ... (A254642) Fourth partial sums: 1, 260, 7595, 94360, 723534, ... (this sequence) MAPLE 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 MATHEMATICA 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 *) Accumulate[Accumulate[Accumulate[Accumulate[Range[20]^8]]]] (* or *) CoefficientList[Series[(1 +247x +4293x^2 +15619x^3 +15619x^4 +4293x^5 + 247x^6 +x^7)/(1-x)^13, {x, 0, 19}], x] PROG (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 (PARI) vector(30, n, m=n+2; binomial(m+2, 5)*m*(m^2-3)*(2*m^4-28*m^2 +101)/198) (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 (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 (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 CROSSREFS Cf. A000542, A001016, A253636, A254642, A254644, A254645, A254646. Sequence in context: A287923 A238029 A264254 * A168187 A067639 A264162 Adjacent sequences:  A254644 A254645 A254646 * A254648 A254649 A254650 KEYWORD nonn,easy AUTHOR Luciano Ancora, Feb 05 2015 STATUS approved

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Last modified May 15 18:17 EDT 2021. Contains 343920 sequences. (Running on oeis4.)