A233440 -id:A233440 - cp's OEIS frontend

A027767 a(n) = (n+1)*binomial(n+1,7).

7, 64, 324, 1200, 3630, 9504, 22308, 48048, 96525, 183040, 330616, 572832, 957372, 1550400, 2441880, 3751968, 5638611, 8306496, 12017500, 17102800, 23976810, 33153120, 45262620, 61074000, 81516825, 107707392, 140977584, 182906944, 235358200, 300516480, 380932464

Offset: 6

Thi Ngoc Dinh (via R. K. Guy)

Number of 9-subsequences of [ 1, n ] with just 1 contiguous pair.

229*a(n) is the number of permutations of (n+1) symbols that 7-commute with an (n+1)-cycle (see A233440 for definition), where 229 = A000757(7). - Luis Manuel Rivera Martínez, Feb 07 2014

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000757, A233440.

Table[(n+1)Binomial[n+1,7],{n,6,40}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{7,64,324,1200,3630,9504,22308,48048,96525},30] (* Harvey P. Dale, Mar 13 2016 *)

G.f.: (7+x)*x^6/(1-x)^9.
From Amiram Eldar, Jan 30 2022: (Start)
Sum_{n>=6} 1/a(n) = 7*Pi^2/6 - 6811/600.
Sum_{n>=6} (-1)^n/a(n) = 7*Pi^2/12 + 2912*log(2)/15 - 252343/1800. (End)

Incorrect formula deleted by R. J. Mathar, Feb 13 2016

A027768 a(n) = (n+1)*binomial(n+1,8).

Original entry on oeis.org

8, 81, 450, 1815, 5940, 16731, 42042, 96525, 205920, 413270, 787644, 1436058, 2519400, 4273290, 7034940, 11277222, 17651304, 27039375, 40619150, 59942025, 87026940, 124472205, 175587750, 244550475, 336585600, 458177148, 617310936, 823753700, 1089372240

Offset: 7

Thi Ngoc Dinh (via R. K. Guy)

Number of 10-subsequences of [ 1, n ] with just 1 contiguous pair.

1625*a(n) is the number of permutations of (n+1) symbols that 8-commute with an (n+1)-cycle (see A233440 for definition), where 1625 = A000757(8). - Luis Manuel Rivera Martínez, Feb 07 2014

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 (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A000757, A233440.

Table[(n+1)Binomial[n+1,8],{n,7,40}] (* Harvey P. Dale, Jul 08 2017 *)

a(n) = (n+1)*binomial(n+1, 8); \\ Michel Marcus, Jan 31 2014

G.f.: (8+x)*x^7/(1-x)^10.
From Amiram Eldar, Jan 30 2022: (Start)
Sum_{n>=7} 1/a(n) = 48877/3675 - 4*Pi^2/3.
Sum_{n>=7} (-1)^(n+1)/a(n) = 2*Pi^2/3 + 38656*log(2)/105 - 2884681/11025. (End)

Incorrect formula deleted . - R. J. Mathar, Feb 13 2016

cp's OEIS Frontend

A027767 a(n) = (n+1)*binomial(n+1,7).

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