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%I #16 Oct 12 2022 09:34:36
%S 1,8,43,211,1030,5055,24851,122166,600470,2951330,14505951,71297834,
%T 350434385,1722411860,8465785506,41609980404,204516223418,
%U 1005212819668,4940697593195,24283905085013,119357243593561,586649945651116
%N (7*n+6)-th terms of expansion of 1/(1-x-x^7).
%H G. C. Greubel, <a href="/A099253/b099253.txt">Table of n, a(n) for n = 0..500</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_07">Index entries for linear recurrences with constant coefficients</a>, signature (8,-21,35,-35,21,-7,1).
%F G.f.: 1/((1-x)^7 - x);
%F Equals A099239(n, 7).
%F 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).
%F a(n) = Sum_{k=0..n} binomial(7*n - 6*(k-1), k).
%F a(n) = Sum_{k=0..n} binomial(n + 6*(k+1), k + 6*(k+1)).
%F a(n) = Sum_{k=0..n} binomial(n + 6*(k+1), n-k).
%t Table[Sum[Binomial[7*n-6*(j-1), j], {j,0,n}], {n,0,30}] (* _G. C. Greubel_, Mar 09 2021 *)
%o (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
%o (Magma) [(&+[Binomial(7*n-6*j+6,j): j in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Mar 09 2021
%Y Cf. A099239.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Oct 08 2004
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