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%I #60 Oct 24 2022 22:05:48
%S 1,15,111,561,2211,7293,21021,54483,129558,286858,598026,1184118,
%T 2242266,4083366,7184166,12257850,20348031,32951985,52179985,80958735,
%U 123288165,184562235,271965915,394962165,565884540,800652996,1119632580,1548656956
%N Seventh partial sums of cubes (A000578).
%H Luciano Ancora, <a href="/A254869/b254869.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_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 + 4*x + x^2)/(1 - x)^11.
%F a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 7*n + n^2)/604800.
%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) + n^3.
%F Sum_{n>=1} 1/a(n) = 1920*sqrt(3/7)*Pi*tan(sqrt(21)*Pi/2) - 251488/49. - _Amiram Eldar_, Jan 26 2022
%e 2nd differences: 0, 6, 12, 18, 24, 30, ... (A008588)
%e 1st differences: 1, 7, 19, 37, 61, 91, ... (A003215)
%e -------------------------------------------------------------------
%e The cubes: 1, 8, 27, 64, 125, 216, ... (A000578)
%e -------------------------------------------------------------------
%e 1st partial sums: 1, 9, 36, 100, 225, 441, ... (A000537)
%e 2nd partial sums: 1, 10, 46, 146, 371, 812, ... (A024166)
%e 3rd partial sums: 1, 11, 57, 203, 574, 1386, ... (A101094)
%e 4th partial sums: 1, 12, 69, 272, 846, 2232, ... (A101097)
%e 5th partial sums: 1, 13, 82, 354, 1200, 3432, ... (A101102)
%e 6th partial sums: 1, 14, 96, 450, 1650, 5082, ... (A254469)
%e 7th partial sums: 1, 15, 111, 561, 2211, 7293, ... (this sequence)
%t Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (7 + n) (7 + 7 n + n^2)/604800, {n, 26}] (* or *)
%t CoefficientList[Series[(- 1 - 4 x - x^2)/(- 1 + x)^11, {x, 0, 25}], x]
%t Nest[Accumulate,Range[30]^3,7] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,15,111,561,2211,7293,21021,54483,129558,286858,598026},30] (* _Harvey P. Dale_, Apr 24 2017 *)
%o (PARI) vector(50, n, n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 7*n + n^2)/604800) \\ _Derek Orr_, Feb 19 2015
%o (Magma) [n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n)*(7+n)*(7+7*n+n^2)/604800: n in [1..30]]; // _Vincenzo Librandi_, Feb 19 2015
%Y Cf. A000537, A000578, A003215, A024166, A101094, A101097, A101102, A254469, A254870, A254871, A254872.
%K nonn,easy
%O 1,2
%A _Luciano Ancora_, Feb 17 2015
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