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 A101094 #53 May 27 2025 14:59:37
%S A101094 1,11,57,203,574,1386,2982,5874,10791,18733,31031,49413,76076,113764,
%T A101094 165852,236436,330429,453663,612997,816431,1073226,1394030,1791010,
%U A101094 2277990,2870595,3586401,4445091,5468617,6681368,8110344,9785336
%N A101094 a(n) = n*(n+1)*(n+2)*(n+3)*(1+3*n+n^2)/120.
%C A101094 Partial sums of A024166. Third partial sums of cubes (A000578).
%C A101094 Antidiagonal sums of the array A213564. - _Clark Kimberling_, Jun 18 2012
%H A101094 Danny Rorabaugh, <a href="/A101094/b101094.txt">Table of n, a(n) for n = 1..10000</a>
%H A101094 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 A101094 C. Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a> [Dead link]
%H A101094 C. Rossiter, <a href="/A101094/a101094_1.pdf">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a> [Cached copy, May 15 2013]
%H A101094 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A101094 This sequence could be obtained from the general formula n*(n+1)*(n+2)*(n+3)*...*(n+k)*(n*(n+k)+(k-1)*k/6)/((k+3)!/6) at k=3. - _Alexander R. Povolotsky_, May 17 2008
%F A101094 G.f.: -x*(1+4*x+x^2) / (x-1)^7. - _R. J. Mathar_, Dec 06 2011
%F A101094 Sum_{n>0} 1/a(n) = (8/3)*(25-9*sqrt(5)*Pi*tan(sqrt(5)*Pi/2)). - _Enrique Pérez Herrero_, Dec 02 2014
%F A101094 a(k) = MagicNKZ(3,k,7) where MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n+1-z,j)*(k-j+1)^n. (Cf. A101104.) - _Danny Rorabaugh_, Apr 23 2015
%t A101094 Table[n*(n + 1)*(n + 2)*(n + 3)*(1 + 3*n + n^2)/120, {n, 31}] (* _Michael De Vlieger_, Apr 20 2015 *)
%o A101094 (Sage) [n*(n+1)*(n+2)*(n+3)*(1+3*n+n^2)/120 for n in range(1,32)] # _Danny Rorabaugh_, Apr 20 2015
%o A101094 (Magma) [n*(n+1)*(n+2)*(n+3)*(1+3*n+n^2)/120 : n in [1..35]]; // _Vincenzo Librandi_, Apr 23 2015
%Y A101094 Cf. A000537, A101097, A101102, A101104.
%K A101094 nonn,easy
%O A101094 1,2
%A A101094 Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
%E A101094 Edited by _Ralf Stephan_, Dec 16 2004