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%I #53 Sep 08 2022 08:46:11
%S 1,68,995,7672,40614,166992,571626,1701480,4534959,11050468,24997973,
%T 53113424,106959580,205628736,379603812,676144944,1166649837,
%U 1956528420,3198236503,5108229896,7988730530,12255340240
%N Fourth partial sums of sixth powers (A001014).
%H Luciano Ancora, <a href="/A254645/b254645.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="/A254645/a254645_2.pdf"> Pascal’s triangle and recurrence relations for partial sums of m-th powers </a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F G.f.: x*(1 + 57*x + 302*x^2 + 302*x^3 + 57*x^4 + x^5)/(1 - x)^11.
%F a(n) = n*(1 + n)*(2 + n)^2*(3 + n)*(4 + n)*(- 1 - 8*n + 14*n^2 + 8*n^3 + n^4)/5040.
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + n^6.
%e First differences: 1, 63, 665, 3367, 11529, 31031, ... (A022522)
%e --------------------------------------------------------------------------
%e The sixth powers: 1, 64, 729, 4096, 15625, 46656, ... (A001014)
%e --------------------------------------------------------------------------
%e First partial sums: 1, 65, 794, 4890, 20515, 67171, ... (A000540)
%e Second partial sums: 1, 66, 860, 5750, 26265, 93436, ... (A101093)
%e Third partial sums: 1, 67, 927, 6677, 32942, 126378, ... (A101099)
%e Fourth partial sums: 1, 68, 995, 7672, 40614, 166992, ... (this sequence)
%p seq(binomial(n+4,5)*(n+2)*((n^2+4*n-1)^2-2)/42, n=1..30); # _G. C. Greubel_, Aug 28 2019
%t Table[n (1 + n) (2 + n)^2 (3 + n) (4 + n) (- 1 - 8 n + 14 n^2 + 8 n^3 + n^4)/5040, {n, 22}] (* or *)
%t Accumulate[Accumulate[Accumulate[Accumulate[Range[22]^6]]]] (* or *)
%t CoefficientList[Series[(- 1 - 57 x - 302 x^2 - 302 x^3 - 57 x^4 - x^5)/(- 1 + x)^11, {x, 0, 21}], x]
%t Nest[Accumulate,Range[30]^6,4] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,68,995,7672,40614,166992,571626,1701480,4534959,11050468,24997973},30] (* _Harvey P. Dale_, Dec 27 2015 *)
%o (PARI) vector(30, n, binomial(n+4,5)*(n+2)*((n^2+4*n-1)^2-2)/42) \\ _G. C. Greubel_, Aug 28 2019
%o (Magma) [Binomial(n+4,5)*(n+2)*((n^2+4*n-1)^2-2)/42: n in [1..30]]; // _G. C. Greubel_, Aug 28 2019
%o (Sage) [binomial(n+4,5)*(n+2)*((n^2+4*n-1)^2-2)/42 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+4*n-1)^2-2)/42); # _G. C. Greubel_, Aug 28 2019
%Y Cf. A000540, A001014, A022522, A101093, A101099.
%Y Cf. A254644 (fourth partial sums of fifth powers), A254646 (fourth partial sums of seventh powers).
%K nonn,easy
%O 1,2
%A _Luciano Ancora_, Feb 05 2015
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