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a(n) = Sum_{k=0..n} binomial(k, 8*(n-k)).
2

%I #20 Jun 21 2021 03:01:20

%S 1,1,1,1,1,1,1,1,1,2,10,46,166,496,1288,3004,6436,12871,24312,43776,

%T 75736,126940,208336,340120,564928,980629,1817047,3605252,7531836,

%U 16146326,34716826,73737316,153430156,311652271,617594122,1195477615,2266064352,4221317464

%N a(n) = Sum_{k=0..n} binomial(k, 8*(n-k)).

%H Seiichi Manyama, <a href="/A306752/b306752.txt">Table of n, a(n) for n = 0..3528</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1,1).

%F G.f.: (1-x)^7/((1-x)^8 - x^9).

%F a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) + a(n-9) for n > 8.

%F a(n) = A017867(8*n).

%t a[n_] := Sum[Binomial[k, 8*(n-k)], {k, 0, n}]; Array[a, 38, 0] (* _Amiram Eldar_, Jun 21 2021 *)

%o (PARI) {a(n) = sum(k=0, n, binomial(k, 8*(n-k)))}

%o (PARI) N=66; x='x+O('x^N); Vec((1-x)^7/((1-x)^8-x^9))

%Y Column 8 of A306680.

%Y Cf. A017867.

%K nonn

%O 0,10

%A _Seiichi Manyama_, Mar 07 2019