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G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^3 / (k*x^(2*k)) ).
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%I #6 Jun 03 2023 14:22:30

%S 1,1,1,3,9,25,88,292,1031,3685,13433,49608,185465,699963,2664650,

%T 10217130,39428179,153009240,596761737,2337875430,9195732624,

%U 36301739221,143780858517,571191310205,2275409450019,9087376470138,36377539265376,145937953205705,586645566919856

%N G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^3 / (k*x^(2*k)) ).

%t nmax = 29; A[_] = 0; Do[A[x_] = x + x^2 Exp[Sum[(-1)^(k + 1) A[x^k]^3/(k x^(2 k)), {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

%t a[1] = a[2] = 1; f[n_] := f[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; g[n_] := g[n] = Sum[a[k] f[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[(-1)^(k/d + 1) d g[d + 2], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 29}]

%Y Cf. A007560, A052755, A363388, A363465, A363468.

%K nonn

%O 1,4

%A _Ilya Gutkovskiy_, Jun 03 2023