A254683 - OEIS

%I #26 Jun 13 2015 00:55:24

%S 1,69,1064,8736,49350,216342,787968,2489448,7024407,18074875,43072848,

%T 96186272,203145852,408774588,788378400,1464523344,2631173181,

%U 4587701601,7785938104,12894168000,20882898530,33138238770

%N Fifth partial sums of sixth powers (A001014).

%H Luciano Ancora, <a href="/A254683/b254683.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 + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)/(- 1 + x)^12.

%F a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(5 + 2*n)*(- 3 + 5*n + n^2)*(4 + 15*n + 3*n^2)/332640.

%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + n^6.

%e First differences:   1, 63,  665, 3367, 11529, ...  (A022522)

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

%e The sixth powers:    1, 64,  729, 4096, 15625, ...  (A001014)

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

%e First partial sums:  1, 65,  794, 4890, 20515, ...  (A000540)

%e Second partial sums: 1, 66,  860, 5750, 26265, ...  (A101093)

%e Third partial sums:  1, 67,  927, 6677, 32942, ...  (A254640)

%e Fourth partial sums: 1, 68,  995, 7672, 40614, ...  (A254645)

%e Fifth partial sums:  1, 69, 1064, 8736, 49350, ...  (this sequence)

%t Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (5 + 2*n) (- 3 + 5*n + n^2) (4 + 15 n + 3 n^2)/332640, {n,22}] (* or *)

%t CoefficientList[Series[(1 + 57 x + 302 x^2 + 302 x^3 + 57 x^4 + x^5)/(- 1 + x)^12, {x,0,21}], x]

%Y Cf. A000540, A001014, A022522, A101093, A254640, A254645, A254681, A254682, A254684.

%K nonn,easy

%O 1,2

%A _Luciano Ancora_, Feb 12 2015