A027764 a(n) = (n+1)*binomial(n+1,4).

4, 25, 90, 245, 560, 1134, 2100, 3630, 5940, 9295, 14014, 20475, 29120, 40460, 55080, 73644, 96900, 125685, 160930, 203665, 255024, 316250, 388700, 473850, 573300, 688779, 822150, 975415, 1150720, 1350360, 1576784, 1832600, 2120580, 2443665, 2804970, 3207789

Offset: 3

Author

Thi Ngoc Dinh (via R. K. Guy)

Keywords

• nonn
• easy

Comments

Number of 6-subsequences of [ 1, n ] with just 1 contiguous pair.

a(n) is also the number of permutations of n+1 symbols that 4-commute with an (n+1)-cycle (see A233440 for definition). - Luis Manuel Rivera Martínez, Feb 07 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 3..1000
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A233440.

Programs

Magma

[(n+1)*Binomial(n+1,4): n in [3..35]]; // Vincenzo Librandi, Feb 08 2014

Mathematica

Table[(n + 1)Binomial[n + 1, 4], {n, 3, 40}] (* or *) LinearRecurrence[ {6, -15, 20, -15, 6, -1}, {4, 25, 90, 245, 560, 1134}, 40] (* Harvey P. Dale, Jun 14 2013 *)
CoefficientList[Series[(4 + x)/(1 - x)^6, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 08 2014 *)

Formula

G.f.: (4+x)*x^3/(1-x)^6.

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). - Harvey P. Dale, Jun 14 2013

a(n) = 10*C(n+2,2)*C(n+2,5)/(n+2)^2. - Gary Detlefs, Aug 20 2013

Sum_{n>=3} 1/a(n) = 62/9 - (2/3)*Pi^2. - Jaume Oliver Lafont, Jul 15 2017

Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/3 + 80*log(2)/3 - 194/9. - Amiram Eldar, Jan 28 2022