%I #28 Sep 08 2022 08:46:11
%S 1,131,2577,23723,141694,636426,2331462,7323954,20396871,51550213,
%T 120271151,262391493,540659756,1060489444,1992739932,3605846676,
%U 6310148349,10717864983,17722868317,28605158351,45165823626,69899222030,106210179010,158685165990
%N Third partial sums of seventh powers (A001015).
%H Luciano Ancora, <a href="/A254641/b254641.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 <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 +120*x +1191*x^2 +2416*x^3 +1191*x^4 +120*x^5 +x^6)/(1-x)^11.
%F a(n) = n*(1+n)*(2+n)*(3+n)*(6 -6*n -20*n^2 +15*n^3 +25*n^4 +9*n^5 +n^6)/720.
%F E.g.f.: x (720 +46440*x +262440*x^2 +425910*x^3 +285264*x^4 +92526*x^5 +15600*x^6 +1380*x^7 +60*x^8 +x^9)*exp(x)/6!. - _G. C. Greubel_, Aug 28 2019
%p seq(binomial(n+3,4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2-6*n+6)/30, n=1..30); # _G. C. Greubel_, Aug 28 2019
%t Table[n(1+n)(2+n)(3+n)(6 -6n -20n^2 +15n^3 +25n^4 +9n^5 +n^6)/720, {n, 30}]
%t CoefficientList[Series[(1 +120x +1191x^2 +2416x^3 +1191x^4 +120x^5 + x^6)/(1-x)^11, {x, 0, 30}], x]
%t Nest[Accumulate,Range[30]^7,3] (* or *) LinearRecurrence[{11,-55,165, -330,462,-462,330,-165,55,-11,1},{1,131,2577,23723, 141694, 636426, 2331462, 7323954,20396871,51550213,120271151},30] (* _Harvey P. Dale_, Jun 19 2018 *)
%o (PARI) Vec((1 +120*x +1191*x^2 +2416*x^3 +1191*x^4 +120*x^5 +x^6)/(1-x)^11 + O(x^40)) \\ _Andrew Howroyd_, Nov 06 2018
%o (PARI) vector(30, n, binomial(n+3,4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2 -6*n+6)/30) \\ _G. C. Greubel_, Aug 28 2019
%o (Magma) [Binomial(n+3,4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2-6*n+6)/30: n in [1..30]]; // _G. C. Greubel_, Aug 28 2019
%o (Sage) [binomial(n+3,4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2-6*n+6)/30 for n in (1..30)] # _G. C. Greubel_, Aug 28 2019
%o (GAP) List([1..30], n-> Binomial(n+3,4)*(n^6+9*n^5+25*n^4+15*n^3-20*n^2 -6*n+6)/30); # _G. C. Greubel_, Aug 28 2019
%Y Cf. A254640, A254642, A254643.
%K nonn,easy
%O 1,2
%A _Luciano Ancora_, Feb 05 2015