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 A151974 #35 Feb 16 2025 08:33:09
%S A151974 0,15,90,315,840,1890,3780,6930,11880,19305,30030,45045,65520,92820,
%T A151974 128520,174420,232560,305235,395010,504735,637560,796950,986700,
%U A151974 1210950,1474200,1781325,2137590,2548665,3020640,3560040,4173840,4869480,5654880,6538455,7529130
%N A151974 a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)/8.
%C A151974 Also the number of 4-cycles in the (n+3)-triangular graph. - _Eric W. Weisstein_, Aug 14 2017
%H A151974 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.
%H A151974 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JohnsonGraph.html">Johnson Graph</a>.
%H A151974 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>.
%H A151974 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A151974 a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)/8.
%F A151974 G.f.: 15*x/(1-x)^6. - _Colin Barker_, Jun 25 2012
%F A151974 a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - _Eric W. Weisstein_, Aug 14 2017
%F A151974 From _Amiram Eldar_, Jan 09 2022: (Start)
%F A151974 Sum_{n>=1} 1/a(n) = 1/12.
%F A151974 Sum_{n>=1} (-1)^(n+1)/a(n) = 16*log(2)/3 - 131/36. (End)
%p A151974 A151974:=n->n*(n+1)*(n+2)*(n+3)*(n+4)/8: seq(A151974(n), n=0..60); # _Wesley Ivan Hurt_, Feb 11 2017
%t A151974 Table[Pochhammer[n, 5]/8, {n, 0, 31}] (* or *)
%t A151974 Rest @ CoefficientList[Series[15 x^2/(1 - x)^6, {x, 0, 32}], x] (* _Michael De Vlieger_, Feb 12 2017 *)
%t A151974 Pochhammer[Range[0, 20], 5]/8 (* _Eric W. Weisstein_, Aug 14 2017 *)
%t A151974 LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 15, 90, 315, 840, 1890}, 20] (* _Eric W. Weisstein_, Aug 14 2017 *)
%t A151974 Table[15 Binomial[n + 4, 5], {n, 0, 20}] (* _Eric W. Weisstein_, Aug 14 2017 *)
%t A151974 15 Binomial[Range[4, 24], 5] (* _Eric W. Weisstein_, Aug 14 2017 *)
%t A151974 Table[(24 n+50 n^2+35 n^3+10 n^4+n^5)/8,{n,0,40}] (* or *) Table[Times@@Range[n,n+4]/8,{n,0,40}] (* _Harvey P. Dale_, Mar 06 2024 *)
%o A151974 (PARI) a(n)=n*(n+1)*(n+2)*(n+3)*(n+4)/8 \\ _Charles R Greathouse IV_, Aug 14 2017
%Y A151974 Cf. A054559.
%Y A151974 Cf. A002417 (number of 3-cycles in the triangular graph), A290939 (5-cycles), A290940 (6-cycles).
%K A151974 nonn,easy
%O A151974 0,2
%A A151974 _Vladimir Joseph Stephan Orlovsky_, Jul 21 2009
%E A151974 Offset corrected by _Eric W. Weisstein_, Aug 14 2017