%I #34 Sep 08 2022 08:45:16

%S 1,12,69,272,846,2232,5214,11088,21879,40612,71643,121056,197132,

%T 310896,476748,713184,1043613,1497276,2110273,2926704,3999930,5393960,

%U 7184970,9462960,12333555,15919956,20365047,25833664,32515032,40625376,50410712

%N a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(2 + 4*n + n^2)/840.

%C Fourth partial sums of cubes (A000578). Partial sums of A101094.

%H G. C. Greubel, <a href="/A101097/b101097.txt">Table of n, a(n) for n = 1..5000</a>

%H C. Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>. [broken link]

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).

%F a(n) = n*(n+1)*(n+2)*(n+3)*...*(n+k)*(n*(n+k) + (k-1)*k/6)/((k+3)!/6) for k=4. - _Alexander R. Povolotsky_, May 17 2008

%F G.f.: x*(1 + 4*x + x^2)/(1-x)^8. - _R. J. Mathar_, Jun 13 2008

%F a(n) = Sum_{k=1..n} A000217(k)^2*A000217(n-k+1). - _Bruno Berselli_, Sep 04 2013

%F E.g.f.: x*(840 + 4200*x + 5040*x^2 + 2240*x^3 + 427*x^4 + 35*x^5 + x^6) *exp(x)/840. - _G. C. Greubel_, Dec 01 2018

%t Table[Binomial[n+4,5]*(2+4*n+n^2)/7, {n,0,50}] (* _G. C. Greubel_, Feb 17 2017 *)

%o (PARI) {A101097(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(2+4*n+n^2)/840} \\ _R. J. Mathar_, Dec 06 2011

%o (Magma) A000217:=func<i | i*(i+1)/2>; [&+[A000217(k)^2*A000217(n-k+1): k in [1..n]]: n in [1..40]]; // _Bruno Berselli_, Sep 04 2013

%o (Sage) [binomial(n+4,5)*(2+4*n+n^2)/7 for n in (1..40)] # _G. C. Greubel_, Dec 01 2018

%Y Cf. A000217, A101102, A101094, A024166, A000537.

%K nonn,easy

%O 1,2

%A Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

%E Edited by _Ralf Stephan_, Dec 16 2004