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a(n) = 1 + Sum_{k=0..n-5} binomial(n-4,k) * a(k).
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%I #5 May 21 2021 08:07:01

%S 1,1,1,1,1,2,4,8,16,32,70,170,452,1277,3731,11145,34031,106888,348016,

%T 1180538,4173726,15320402,58053312,225891952,899492200,3660479037,

%U 15228099789,64831944993,282763031581,1263953233142,5788015999020,27121892020940,129849269955372,634208223729772

%N a(n) = 1 + Sum_{k=0..n-5} binomial(n-4,k) * a(k).

%F G.f. A(x) satisfies: A(x) = (1 + x^4 * A(x/(1 - x))) / ((1 - x) * (1 + x^4)).

%t a[n_] := a[n] = 1 + Sum[Binomial[n - 4, k] a[k] , {k, 0, n - 5}]; Table[a[n], {n, 0, 33}]

%t nmax = 33; A[_] = 0; Do[A[x_] = (1 + x^4 A[x/(1 - x)])/((1 - x) (1 + x^4)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%Y Cf. A000629, A210542, A344489, A344490, A344491, A344493.

%K nonn

%O 0,6

%A _Ilya Gutkovskiy_, May 21 2021