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A352045 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * a(k). 5

%I #9 May 30 2022 09:55:18

%S 1,1,1,1,2,5,11,22,43,85,170,341,683,1367,2744,5552,11381,23729,50423,

%T 109220,240845,539411,1223357,2800535,6451340,14915084,34533851,

%U 79949741,184865936,426627122,982217705,2255539406,5166125852,11802977573,26903285366

%N a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * a(k).

%H Seiichi Manyama, <a href="/A352045/b352045.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f. A(x) satisfies: A(x) = 1 + x * A(x^3/(1 - x)^3) / (1 - x).

%F E.g.f.: Integral exp(x) * Sum_{n>=0} a(n) * x^(3*n) / (3*n)! dx.

%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 3 k] a[k], {k, 0, Floor[(n - 1)/3]}]; Table[a[n], {n, 0, 34}]

%t nmax = 34; A[_] = 0; Do[A[x_] = 1 + x A[x^3/(1 - x)^3]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%Y Cf. A351970, A352044.

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, Mar 01 2022

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