A345233 G.f. A(x) satisfies: A(x) = x + x^2 / exp(A(x) - A(x^2)/2 + A(x^3)/3 - A(x^4)/4 + ...).
1, 1, -1, 0, 1, 0, -2, 1, 3, -4, -3, 11, -2, -22, 21, 32, -72, -18, 180, -95, -350, 496, 449, -1542, 125, 3638, -3161, -6393, 12780, 5636, -35993, 14509, 77907, -97880, -116880, 337924, 24514, -869531, 631306, 1692540, -2949009, -1933940, 9035577, -2312868, -21166895
Offset: 1
Keywords
Programs
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Mathematica
nmax = 45; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[(-1)^k 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] = (1/(n - 2)) Sum[Sum[(-1)^(k/d) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 45}]
Formula
G.f.: x + x^2 / Product_{n>=1} (1 + x^n)^a(n).
a(n+2) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d) * d * a(d) ) * a(n-k+2).