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Sixth partial sums of cubes (A000578).
8

%I #43 Jan 26 2022 02:29:34

%S 1,14,96,450,1650,5082,13728,33462,75075,157300,311168,586092,1058148,

%T 1841100,3100800,5073684,8090181,12603954,19228000,28778750,42329430,

%U 61274070,87403680,122996250,170922375,234768456,318979584,429024376,571584200,754769400

%N Sixth partial sums of cubes (A000578).

%H Luciano Ancora, <a href="/A254469/b254469.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_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

%F G.f.: (x + 4*x^2 + x^3)/(- 1 + x)^10.

%F a(n) = n*(1 + n)^2*(2 + n)*(3 + n)*(4 + n)*(5 + n)^2*(6 + n)/60480.

%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + n^3.

%F From _Amiram Eldar_, Jan 26 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 217/200.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 223769/200 - 8064*log(2)/5. (End)

%e First differences: 1, 7, 19, 37, 61, 91, ... (A003215)

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

%e The cubes: 1, 8, 27, 64, 125, 216, ... (A000578)

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

%e First partial sums: 1, 9, 36, 100, 225, 441, ... (A000537)

%e Second partial sums: 1, 10, 46, 146, 371, 812, ... (A024166)

%e Third partial sums: 1, 11, 57, 203, 574, 1386, ... (A101094)

%e Fourth partial sums: 1, 12, 69, 272, 846, 2232, ... (A101097)

%e Fifth partial sums: 1, 13, 82, 354, 1200, 3432, ... (A101102)

%e Sixth partial sums: 1, 14, 96, 450, 1650, 5082, ... (this sequence)

%t Table[n (1 + n)^2 (2 + n) (3 + n) (4 + n) (5 + n)^2 (6 + n)/60480, {n, 27}] (* or *) CoefficientList[Series[(1 + 4 x + x^2)/(- 1 + x)^10, {x, 0, 26}], x]

%t Nest[Accumulate,Range[30]^3,6] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,14,96,450,1650,5082,13728,33462,75075,157300},30] (* _Harvey P. Dale_, Sep 03 2016 *)

%o (Magma) [n*(1+n)^2*(2+n)*(3+n)*(4+n)*(5+n)^2*(6+n)/60480: n in [1..30]]; // _Vincenzo Librandi_, Feb 15 2015

%o (PARI) a(n)=n*(1+n)^2*(2+n)*(3+n)*(4+n)*(5+n)^2*(6+n)/60480 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A000537, A000578, A003215, A024166, A101094, A101097, A101102, A254470, A254471, A254472.

%K nonn,easy

%O 1,2

%A _Luciano Ancora_, Feb 15 2015