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A363087
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 + ...).
0
1, -1, -1, 1, 2, -1, -5, -1, 11, 10, -21, -39, 30, 126, 4, -354, -261, 834, 1347, -1483, -5033, 823, 15663, 8765, -41112, -56364, 84888, 234546, -91319, -791833, -293380, 2251507, 2561264, -5177875, -11835968, 7620048, 42944358, 7464956, -130615874, -119900209
OFFSET
1,5
FORMULA
G.f.: x - x^2 * Product_{n>=1} (1 + x^n)^a(n).
a(1) = 1, a(2) = -1; a(n) = (1/(n - 2)) * Sum_{k=1..n-2} ( Sum_{d|k} (-1)^(k/d+1) * d * a(d) ) * a(n-k).
MATHEMATICA
nmax = 40; A[_] = 0; Do[A[x_] = x - x^2 Exp[Sum[(-1)^(k + 1) A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 1; a[2] = -1; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[(-1)^(k/d + 1) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 40}]
CROSSREFS
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, May 18 2023
STATUS
approved