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a(n) = Sum_{k=0..n} binomial(5*k,n-k).
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%I #13 Jan 25 2023 09:07:51

%S 1,1,6,21,71,251,882,3088,10829,37975,133146,466852,1636944,5739647,

%T 20125051,70564951,247423522,867546829,3041899638,10665883415,

%U 37398034921,131129599227,459782762029,1612146986543,5652708454881,19820223058176,69496108849357

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

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,10,10,5,1).

%F a(n) = a(n-1) + 5*a(n-2) + 10*a(n-3) + 10*a(n-4) + 5*a(n-5) + a(n-6).

%F G.f.: 1/(1 - x*(1+x)^5).

%o (PARI) a(n) = sum(k=0, n, binomial(5*k, n-k));

%o (PARI) my(N=30, x='x+O('x^N)); Vec(1/(1-x*(1+x)^5))

%Y Cf. A002478, A099234, A099235.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Jan 25 2023