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A307365
G.f. A(x) satisfies: A(x) = x*exp(A(-x) + A(-x^2)/2 + A(-x^3)/3 + A(-x^4)/4 + ...).
3
0, 1, -1, -1, 2, 1, -4, -3, 11, 10, -36, -32, 122, 105, -420, -368, 1497, 1336, -5491, -4919, 20477, 18393, -77397, -69883, 296306, 268711, -1146538, -1042924, 4475265, 4081598, -17600475, -16091719, 69681964, 63845971, -277494594, -254730047, 1110782803, 1021361912
OFFSET
0,5
FORMULA
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} 1/(1 - x^n)^((-1)^n*a(n)).
Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^d*d*a(d) ) * a(n-k+1).
EXAMPLE
G.f.: A(x) = x - x^2 - x^3 + 2*x^4 + x^5 - 4*x^6 - 3*x^7 + 11*x^8 + 10*x^9 - 36*x^10 - 32*x^11 + ...
MATHEMATICA
terms = 37; A[_] = 0; Do[A[x_] = x Exp[Sum[A[-x^k]/k, {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 - x^k)^((-1)^k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 37}]
CROSSREFS
KEYWORD
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AUTHOR
Ilya Gutkovskiy, Apr 05 2019
STATUS
approved