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Seventh partial sums of fourth powers (A000583).
5

%I #31 Sep 08 2022 08:46:11

%S 1,23,221,1355,6239,23465,75803,217373,566150,1361802,3063502,6508450,

%T 13159666,25481470,47493274,85567222,149553199,254336185,421956275,

%U 684451365,1087616985,1695917535,2598828765,3918943275,5822229660,8530902276,12339433068

%N Seventh partial sums of fourth powers (A000583).

%H Luciano Ancora, <a href="/A254870/b254870.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_12">Index entries for linear recurrences with constant coefficients</a>, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

%F G.f.: (x + 11*x^2 + 11*x^3 + x^4)/(- 1 + x)^12.

%F a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 2*n)*(7 + 42*n + 6*n^2)/19958400.

%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^4.

%e Second differences: 2, 14, 50, 110, 194, 302, ... A120328(2k+1)

%e First differences: 1, 15, 65, 175, 369, 671, ... A005917

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

%e The fourth powers: 1, 16, 81, 256, 625, 1296, ... A000583

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

%e First partial sums: 1, 17, 98, 354, 979, 2275, ... A000538

%e Second partial sums: 1, 18, 116, 470, 1449, 3724, ... A101089

%e Third partial sums: 1, 19, 135, 605, 2054, 5778, ... A101090

%e Fourth partial sums: 1, 20, 155, 760, 2814, 8592, ... A101091

%e Fifth partial sums: 1, 21, 176, 936, 3750, 12342, ... A254681

%e Sixth partial sums: 1, 22, 198, 1134, 4884, 17226, ... A254470

%e Seventh partial sums: 1, 23, 221, 1355, 6239, 23465, ... (this sequence)

%t Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (7 + n) (7 + 2 n)((7 + 42 n + 6 n^2)/19958400), {n, 24}] (* or *)

%t CoefficientList[Series[(1 + 11 x + 11 x^2 + x^3)/(- 1 + x)^12, {x, 0, 23}], x]

%o (PARI) vector(50,n,n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 2*n)*(7 + 42*n + 6*n^2)/19958400) \\ _Derek Orr_, Feb 19 2015

%o (Magma) [n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n)*(7+n)*(7+2*n)*(7 +42*n+6*n^2)/19958400: n in [1..30]]; // _Vincenzo Librandi_, Feb 19 2015

%Y Cf. A000538, A000583, A005917, A101089, A101090, A101091, A254681, A254470, A254869, A254871, A254872.

%K nonn,easy

%O 1,2

%A _Luciano Ancora_, Feb 17 2015