A151974 a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)/8.

0, 15, 90, 315, 840, 1890, 3780, 6930, 11880, 19305, 30030, 45045, 65520, 92820, 128520, 174420, 232560, 305235, 395010, 504735, 637560, 796950, 986700, 1210950, 1474200, 1781325, 2137590, 2548665, 3020640, 3560040, 4173840, 4869480, 5654880, 6538455, 7529130

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Jul 21 2009

Keywords

• nonn
• easy

Comments

Also the number of 4-cycles in the (n+3)-triangular graph. - Eric W. Weisstein, Aug 14 2017

Links

• Eric Weisstein's World of Mathematics, Graph Cycle.
• Eric Weisstein's World of Mathematics, Johnson Graph.
• Eric Weisstein's World of Mathematics, Triangular Graph.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A054559.

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

Programs

Maple

A151974:=n->n*(n+1)*(n+2)*(n+3)*(n+4)/8: seq(A151974(n), n=0..60); # Wesley Ivan Hurt, Feb 11 2017

Mathematica

Table[Pochhammer[n, 5]/8, {n, 0, 31}] (* or *)
Rest @ CoefficientList[Series[15 x^2/(1 - x)^6, {x, 0, 32}], x] (* Michael De Vlieger, Feb 12 2017 *)
Pochhammer[Range[0, 20], 5]/8 (* Eric W. Weisstein, Aug 14 2017 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 15, 90, 315, 840, 1890}, 20] (* Eric W. Weisstein, Aug 14 2017 *)
Table[15 Binomial[n + 4, 5], {n, 0, 20}] (* Eric W. Weisstein, Aug 14 2017 *)
15 Binomial[Range[4, 24], 5] (* Eric W. Weisstein, Aug 14 2017 *)
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 *)

PARI

a(n)=n*(n+1)*(n+2)*(n+3)*(n+4)/8 \\ Charles R Greathouse IV, Aug 14 2017

Formula

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

G.f.: 15*x/(1-x)^6. - Colin Barker, Jun 25 2012

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

From Amiram Eldar, Jan 09 2022: (Start)

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

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

Extensions

Offset corrected by Eric W. Weisstein, Aug 14 2017