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a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)/8.
4

%I #34 Mar 06 2024 14:04:05

%S 0,15,90,315,840,1890,3780,6930,11880,19305,30030,45045,65520,92820,

%T 128520,174420,232560,305235,395010,504735,637560,796950,986700,

%U 1210950,1474200,1781325,2137590,2548665,3020640,3560040,4173840,4869480,5654880,6538455,7529130

%N a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)/8.

%C Also the number of 4-cycles in the (n+3)-triangular graph. - _Eric W. Weisstein_, Aug 14 2017

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JohnsonGraph.html">Johnson Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)/8.

%F G.f.: 15*x/(1-x)^6. - _Colin Barker_, Jun 25 2012

%F 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 From _Amiram Eldar_, Jan 09 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 1/12.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 16*log(2)/3 - 131/36. (End)

%p 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 Table[Pochhammer[n, 5]/8, {n, 0, 31}] (* or *)

%t Rest @ CoefficientList[Series[15 x^2/(1 - x)^6, {x, 0, 32}], x] (* _Michael De Vlieger_, Feb 12 2017 *)

%t Pochhammer[Range[0, 20], 5]/8 (* _Eric W. Weisstein_, Aug 14 2017 *)

%t LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 15, 90, 315, 840, 1890}, 20] (* _Eric W. Weisstein_, Aug 14 2017 *)

%t Table[15 Binomial[n + 4, 5], {n, 0, 20}] (* _Eric W. Weisstein_, Aug 14 2017 *)

%t 15 Binomial[Range[4, 24], 5] (* _Eric W. Weisstein_, Aug 14 2017 *)

%t 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 (PARI) a(n)=n*(n+1)*(n+2)*(n+3)*(n+4)/8 \\ _Charles R Greathouse IV_, Aug 14 2017

%Y Cf. A054559.

%Y Cf. A002417 (number of 3-cycles in the triangular graph), A290939 (5-cycles), A290940 (6-cycles).

%K nonn,easy

%O 0,2

%A _Vladimir Joseph Stephan Orlovsky_, Jul 21 2009

%E Offset corrected by _Eric W. Weisstein_, Aug 14 2017