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%I #28 Feb 06 2023 18:59:06
%S 1,13,82,354,1200,3432,8646,19734,41613,82225,153868,274924,472056,
%T 782952,1259700,1972884,3016497,4513773,6624046,9550750,13550680,
%U 18944640,26129610,35592570,47926125,63846081,84211128,110044792,142559824,183185200,233595912
%N Fifth partial sums of cubes (A000578).
%H Colin Barker, <a href="/A101102/b101102.txt">Table of n, a(n) for n = 1..1000</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*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(10 + 3*n*(n+5))/20160.
%F This sequence could be obtained from the general formula a(n) = n*(n+1)*(n+2)*(n+3)*...*(n+k)*(n*(n+k) + (k-1)*k/6)/((k+3)!/6) at k=5. - _Alexander R. Povolotsky_, May 17 2008
%F G.f.: x*(x^2+4*x+1) / (1-x)^9. - _Colin Barker_, Apr 23 2015
%F Sum_{n>=1} 1/a(n) = -162*sqrt(21/5)*Pi*tan(sqrt(35/3)*Pi/2) - 136269/100. - _Amiram Eldar_, Jan 26 2022
%t Table[Binomial[n+5,6]*(3*n^2+15*n+10)/28, {n,1,30}] (* _G. C. Greubel_, Dec 01 2018 *)
%t Nest[Accumulate,Range[40]^3,5] (* _Harvey P. Dale_, Feb 06 2023 *)
%o (PARI) a(n)=sum(t=1,n,sum(s=1,t,sum(l=1,s,sum(j=1,l, sum(m=1, j, sum(i=m*(m+1)/2-m+1, m*(m+1)/2,(2*i-1))))))) \\ _Alexander R. Povolotsky_, May 17 2008
%o (PARI) Vec(-x*(x^2+4*x+1)/(x-1)^9 + O(x^100)) \\ _Colin Barker_, Apr 23 2015
%o (PARI) a(n) = binomial(n+5,6)*(3*n^2+15*n+10)/28 \\ _Charles R Greathouse IV_, Apr 23 2015
%o (Magma) [Binomial(n+5,6)*(3*n^2+15*n+10)/28: n in [1..30]]; // _G. C. Greubel_, Dec 01 2018
%o (Sage) [binomial(n+5,6)*(3*n^2+15*n+10)/28 for n in (1..30)] # _G. C. Greubel_, Dec 01 2018
%Y Partial sums of A101097.
%Y Cf. A000537, A024166, A101094.
%K easy,nonn
%O 1,2
%A Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
%E Edited by _Ralf Stephan_, Dec 16 2004
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