A099253 (7*n+6)-th terms of expansion of 1/(1-x-x^7).

1, 8, 43, 211, 1030, 5055, 24851, 122166, 600470, 2951330, 14505951, 71297834, 350434385, 1722411860, 8465785506, 41609980404, 204516223418, 1005212819668, 4940697593195, 24283905085013, 119357243593561, 586649945651116

Offset: 0

Author

Paul Barry, Oct 08 2004

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 16.
• Index entries for linear recurrences with constant coefficients, signature (8,-21,35,-35,21,-7,1).

Crossrefs

Cf. A099239.

Programs

Magma

[(&+[Binomial(7*n-6*j+6,j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Mar 09 2021

Mathematica

Table[Sum[Binomial[7*n-6*(j-1), j], {j,0,n}], {n,0,30}] (* G. C. Greubel, Mar 09 2021 *)

Sage

[sum(binomial(7*n-6*j+6,j) for j in (0..n)) for n in (0..30)] # G. C. Greubel, Mar 09 2021

Formula

G.f.: 1/((1-x)^7 - x);

Equals A099239(n, 7).

a(n) = 8*a(n-1) -21*a(n-2) +35*a(n-3) -35*a(n-4) +21*a(n-5) -7*a(n-6) +a(n-7).

a(n) = Sum_{k=0..n} binomial(7*n - 6*(k-1), k).

a(n) = Sum_{k=0..n} binomial(n + 6*(k+1), k + 6*(k+1)).

a(n) = Sum_{k=0..n} binomial(n + 6*(k+1), n-k).