A099242 - OEIS

%I #30 Aug 30 2024 10:19:41

%S 1,7,34,153,686,3088,13917,62721,282646,1273690,5739647,25864698,

%T 116554700,525233175,2366870474,10665883415,48063918336,216591552484,

%U 976031547888,4398313653120,19820223058176,89316331907533

%N (6n+5)-th terms of expansion of 1/(1 - x - x^6).

%C A row of A099239.

%C Equals INVERT transform of A000389, C(n,5). [_Gary W. Adamson_, Feb 02 2009]

%H G. C. Greubel, <a href="/A099242/b099242.txt">Table of n, a(n) for n = 0..1000</a>

%H D. Birmajer, J. B. Gil, and M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet</a>, J. Int. Seq. 19 (2016) # 16.1.3, example 16.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (7,-15,20,-15,6,-1).

%F G.f.: 1/((1-x)^6-x).

%F a(n) = 7*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).

%F a(n) = Sum_{k=0..n} binomial(6*n-5*(k-1), k).

%F a(n) = Sum_{k=0..n} binomial(n+5*(k+1), k+5*(k+1)).

%F a(n) = Sum_{k=0..n} binomial(n+5*(k+1), n-k).

%t CoefficientList[Series[1/((1 - x)^6 - x), {x, 0, 50}], x] (* _G. C. Greubel_, Nov 24 2017 *)

%t LinearRecurrence[{7,-15,20,-15,6,-1},{1,7,34,153,686,3088},30] (* _Harvey P. Dale_, May 06 2018 *)

%o (PARI) my(x='x+O('x^50)); Vec(1/((1-x)^6-x)) \\ _G. C. Greubel_, Nov 24 2017

%Y Cf. A000389.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Oct 08 2004