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%I #8 Nov 06 2021 09:06:25
%S 1,1,2,4,9,20,47,111,270,663,1656,4174,10636,27308,70651,183902,
%T 481436,1266515,3346793,8879116,23642034,63156917,169222939,454660940,
%U 1224650739,3306338583,8945780742,24252558183,65872671839,179228552638,488443704486
%N G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x) / (1 - x) + x^2 * A(x)^2.
%F a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} a(k) * (1 + a(n-k-2)).
%F a(n) ~ sqrt(1/r + (2-r)*s/(1-r)^2 + 2*s^2) / (2*sqrt(Pi)*n^(3/2)*r^n), where r = 0.3495518575342322867499973927570340375314361958565... and s = 3.323404276086477625771682790702806844309937221726... are real roots of the system of equations 1 + r + r^2*s*(1/(1-r) + s) = s, r^2*(1/(1-r) + 2*s) = 1. - _Vaclav Kotesovec_, Nov 06 2021
%t nmax = 30; A[_] = 0; Do[A[x_] = 1 + x + x^2 A[x]/(1 - x) + x^2 A[x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t a[0] = a[1] = 1; a[n_] := a[n] = Sum[a[k] (1 + a[n - k - 2]), {k, 0, n - 2}]; Table[a[n], {n, 0, 30}]
%Y Cf. A007477, A215973, A349015, A349016.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 05 2021
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