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proposed

G.f. A(x) satisfies: A(x) = x + x^2 * exp(4 * Sum_{k>=1} A(x^k) / k).

allocated for Ilya Gutkovskiy

G.f. A(x) satisfies: A(x) = x + x^2 * exp(4 * Sum{k>=1} A(x^k) / k).

1, 1, 4, 14, 52, 205, 832, 3492, 14960, 65322, 289384, 1298064, 5882712, 26897352, 123919576, 574718308, 2681028168, 12571650355, 59222213028, 280139215118, 1330101884932, 6336757979653, 30282375754944, 145124083402256, 697293746743760, 3358385599930269, 16210842955175380

1,3

G.f.: x + x^2 / Product_{n>=1} (1 - x^n)^(4*a(n)).

a(n+2) = (4/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d) ) * a(n-k+2).

nmax = 27; A[_] = 0; Do[A[x_] = x + x^2 Exp[4 Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

a[1] = a[2] = 1; a[n_] := a[n] = (4/(n - 2)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 27}]

Cf. A007562, A052763, A345200, A345241.

allocated

nonn

Ilya Gutkovskiy, Jun 11 2021

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editing

allocated for Ilya Gutkovskiy

allocated

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