|
%I #25 Jan 27 2022 03:04:51
%S 1,35,345,1955,7990,26226,73470,182490,412335,863005,1695551,3158805,
%T 5624060,9629140,15933420,25585476,40005165,61082055,91292245,
%U 133835735,192796626,273328550,381867850,526377150,716622075,964484001,1284311835
%N Third partial sums of fifth powers (A000584).
%H G. C. Greubel, <a href="/A101099/b101099.txt">Table of n, a(n) for n = 1..5000</a>
%H Cecilia Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>. [Dead link]
%H Cecilia Rossiter, <a href="/A101096/a101096.pdf">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>. [Cached copy, May 15 2013]
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F a(n) = n*(1 + n)*(2 + n)*(3 + n)*(-1 + n*(2 + n))*(2 + n*(4 + n))/336.
%F G.f.: x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1-x)^9. - _Colin Barker_, Apr 16 2012
%F a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9). - _Harvey P. Dale_, Feb 20 2015
%F E.g.f.: x*(336 + 5544*x + 13608*x^2 + 10934*x^3 + 3696*x^4 + 574*x^5 + 40*x^6 + x^7)*exp(x)/336. - _G. C. Greubel_, Dec 01 2018
%F Sum_{n>=1} 1/a(n) = 224/3 - 60*sqrt(2)*Pi*cot(sqrt(2)*Pi). - _Amiram Eldar_, Jan 27 2022
%t Nest[Accumulate[#]&,Range[30]^5,3] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,35,345,1955,7990,26226,73470,182490,412335},30] (* _Harvey P. Dale_, Feb 20 2015 *)
%o (PARI) vector(30, n, n*(1+n)*(2+n)*(3+n)*(-1+n*(2+n))*(2+n*(4+n))/336) \\ _G. C. Greubel_, Dec 01 2018
%o (Magma) [n*(1+n)*(2+n)*(3+n)*(-1+n*(2+n))*(2+n*(4+n))/336: n in [1..30]]; // _G. C. Greubel_, Dec 01 2018
%o (Sage) [n*(1+n)*(2+n)*(3+n)*(-1+n*(2+n))*(2+n*(4+n))/336 for n in (1..30)] # _G. C. Greubel_, Dec 01 2018
%Y Cf. A000584.
%K nonn,easy
%O 1,2
%A Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
%E Edited by _Ralf Stephan_, Dec 16 2004
|
