This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A085438 #39 Jan 13 2025 11:09:46
%S A085438 1,28,244,1244,4619,13880,35832,82488,173613,339988,627484,1102036,
%T A085438 1855607,3013232,4741232,7256688,10838265,15838476,22697476,31958476,
%U A085438 44284867,60479144,81503720,108503720,142831845,186075396,240085548,307008964,389321839
%N A085438 a(n) = Sum_{i=1..n} binomial(i+1,2)^3.
%D A085438 Elisabeth Busser and Gilles Cohen, Neuro-Logies - "Chercher, jouer, trouver", La Recherche, April 1999, No. 319, page 97.
%H A085438 G. C. Greubel, <a href="/A085438/b085438.txt">Table of n, a(n) for n = 1..5000</a>
%H A085438 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See p. 13.
%H A085438 <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 A085438 a(n) = (90*n^7 +630*n^6 +1638*n^5 +1890*n^4+ 840*n^3 -48*n)/7!.
%F A085438 a(n) = (C(n+2, 3)/35)*(35 +210*C(n-1, 1) +399*C(n-1, 2) +315*C(n-1, 3) +90*C(n-1, 4)).
%F A085438 G.f.: x*(x^4+20*x^3+48*x^2+20*x+1) / (x-1)^8. - _Colin Barker_, May 02 2014
%e A085438 a(10) = (90*(10^7)+630*(10^6)+1638*(10^5)+1890*(10^4)+840*(10^3)-48*(10))/5040 = 339988.
%t A085438 Table[(90*n^7 + 630*n^6 + 1638*n^5 + 1890*n^4 + 840*n^3 - 48*n)/7!, {n, 1, 50}] (* _G. C. Greubel_, Nov 22 2017 *)
%o A085438 (PARI) Vec(x*(x^4+20*x^3+48*x^2+20*x+1)/(x-1)^8 + O(x^100)) \\ _Colin Barker_, May 02 2014
%o A085438 (PARI) a(n) = sum(i=1, n, binomial(i+1, 2)^3); \\ _Michel Marcus_, Nov 22 2017
%o A085438 (Magma) [(90*n^7 +630*n^6 +1638*n^5 +1890*n^4+ 840*n^3 -48*n)/ Factorial(7): n in [1..30]]; // _G. C. Greubel_, Nov 22 2017
%Y A085438 Column k=3 of A334781.
%Y A085438 Cf. A000292, A087127, A024166, A024166, A085439, A085440, A085441, A085442, A000332, A086020, A086021, A086022, A000389, A086023, A086024, A000579, A086025, A086026, A000580, A086027, A086028, A027555, A086029, A086030.
%K A085438 easy,nonn
%O A085438 1,2
%A A085438 _André F. Labossière_, Jun 30 2003
%E A085438 More terms from _Colin Barker_, May 02 2014
%E A085438 Formula and example edited by _Colin Barker_, May 02 2014