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A363386 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^2 / k ). 1

%I #6 May 30 2023 19:39:10

%S 1,1,0,1,2,1,4,8,10,22,50,77,160,343,622,1250,2648,5127,10364,21685,

%T 43594,88907,185458,380113,782902,1633841,3387444,7033401,14716304,

%U 30734066,64228198,134824862,283040684,594516622,1252151812,2639220817,5566237724,11760037378

%N G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^2 / k ).

%t nmax = 38; A[_] = 0; Do[A[x_] = x + x^2 Exp[Sum[(-1)^(k + 1) A[x^k]^2/k, {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

%t a[1] = a[2] = 1; g[n_] := g[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[(-1)^(k/d + 1) d g[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 38}]

%o (PARI) seq(n)=my(p=x+x^2+O(x^3)); for(n=1, n\2, my(m=serprec(p,x)-1); p = x + x^2*exp(-sum(k=1, m\2, (-1)^k*subst(p + O(x^(m\k+1)), x, x^k)^2/k))); Vec(p + O(x*x^n)) \\ _Andrew Howroyd_, May 30 2023

%Y Cf. A005754, A007560, A363385.

%K nonn

%O 1,5

%A _Ilya Gutkovskiy_, May 30 2023

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Last modified February 22 06:21 EST 2024. Contains 370240 sequences. (Running on oeis4.)