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 A085441 #54 Jan 13 2025 11:24:47
%S A085441 1,730,47386,1047386,12438011,98204132,580094436,2756876772,
%T A085441 11060642397,38741283022,121395233038,346594833742,914464085783,
%U A085441 2254559726408,5240543726408,11568062614344,24395756421273,49397866465794,96443747465794,182209868465794
%N A085441 a(n) = Sum_{i=1..n} binomial(i+1,2)^6.
%H A085441 T. D. Noe, <a href="/A085441/b085441.txt">Table of n, a(n) for n = 1..1000</a>
%H A085441 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 A085441 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).
%F A085441 G.f.: x*(x^10 +716*x^9 +37257*x^8 +450048*x^7 +1822014*x^6 +2864328*x^5 +1822014*x^4 +450048*x^3 +37257*x^2 +716*x +1) / (x -1)^14. - _Colin Barker_, May 02 2014
%F A085441 a(n) = (n/960960)*(6112 - 40040*n^2 + 78078*n^4 + 15015*n^5 + 19305*n^6 + 225225*n^7 + 335335*n^8 + 225225*n^9 + 80535*n^10 + 15015*n^11 + 1155*n^12). - _G. C. Greubel_, Nov 22 2017
%e A085441 a(5) = C(7,3)*[191*106 + 450*(18*C(14,10) + 3851*C(13,10) + 61839*C(12,10) + 225352*C(11,10) + 225352*C(10,10))]/10010 = 12438011.
%p A085441 f:= sum(binomial(1+i,2)^6,i=1..n):
%p A085441 seq(f, n=1..30); # _Robert Israel_, Nov 22 2017
%t A085441 Table[Sum[Binomial[i+1,2]^6,{i,n}],{n,20}] (* or *) LinearRecurrence[ {14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1},{1,730,47386,1047386,12438011, 98204132,580094436, 2756876772,11060642397, 38741283022,121395233038, 346594833742, 914464085783, 2254559726408},20] (* _Harvey P. Dale_, Jun 05 2017 *)
%o A085441 (PARI) for(n=1,30, print1(sum(k=1,n, binomial(k+1,2)^6), ", ")) \\ _G. C. Greubel_, Nov 22 2017
%o A085441 (Magma) [(n/960960)*(6112 - 40040*n^2 + 78078*n^4 + 15015*n^5 + 19305*n^6 + 225225*n^7 + 335335*n^8 + 225225*n^9 + 80535*n^10 + 15015*n^11 + 1155*n^12): n in [1..30]]; // _G. C. Greubel_, Nov 22 2017
%Y A085441 Column k=6 of A334781.
%Y A085441 Cf. A000292, A087127, A024166, A024166, A085438, A085439, A085440, A085442, A000332, A086020, A086021, A086022, A000389, A086023, A086024, A000579, A086025, A086026, A000580, A086027, A086028, A027555, A086029, A086030, A234253.
%K A085441 easy,nonn
%O A085441 1,2
%A A085441 _André F. Labossière_, Jul 07 2003