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%I #21 Jan 22 2023 09:15:19
%S 1,1,1,1,2,3,4,5,8,13,20,29,45,73,118,185,293,475,778,1263,2047,3345,
%T 5512,9085,14957,24683,40918,67987,113016,188053,313608,524041,876657,
%U 1467797,2460644,4130893,6942726,11678687,19663068,33139295,55904339,94384167,159470488
%N a(n) = 1 + Sum_{k=0..n-4} a(k) * a(n-k-4).
%H G. C. Greubel, <a href="/A346073/b346073.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^4 * A(x)^2.
%F a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * Catalan(k). - _Seiichi Manyama_, Jan 22 2023
%t a[n_] := a[n] = 1 + Sum[a[k] a[n - k - 4], {k, 0, n - 4}]; Table[a[n], {n, 0, 42}]
%t nmax = 42; A[_] = 0; Do[A[x_] = 1/(1 - x) + x^4 A[x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%o (SageMath)
%o @CachedFunction
%o def a(n): # a = A346073
%o if (n<4): return 1
%o else: return 1 + sum(a(k)*a(n-k-4) for k in range(n-3))
%o [a(n) for n in range(51)] # _G. C. Greubel_, Nov 26 2022
%o (PARI) a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(2*k, k)/(k+1)); \\ _Seiichi Manyama_, Jan 22 2023
%Y Cf. A000108, A007317, A090344, A216604, A307971, A346074.
%K nonn
%O 0,5
%A _Ilya Gutkovskiy_, Jul 04 2021
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