A307366 G.f. A(x) satisfies: A(x) = x*exp(A(-x) - A(-x^2)/2 + A(-x^3)/3 - A(-x^4)/4 + ...).

0, 1, -1, 0, 0, 1, -2, -1, 3, 3, -8, -5, 17, 15, -47, -35, 118, 91, -311, -240, 839, 660, -2314, -1809, 6417, 5035, -18002, -14177, 51016, 40322, -145784, -115402, 419197, 332457, -1212617, -963586, 3526976, 2807301, -10307097, -8215194, 30246994, 24139050, -89101081

Offset: 0

Author

Ilya Gutkovskiy, Apr 05 2019

Keywords

• sign

Examples

			G.f.: A(x) = x - x^2 + x^5 - 2*x^6 - x^7 + 3*x^8 + 3*x^9 - 8*x^10 - 5*x^11 + 17*x^12 + ...
		

Crossrefs

Cf. A000081, A004111, A045648, A049075, A307365.

Programs

Mathematica

terms = 42; A[] = 0; Do[A[x] = x Exp[Sum[(-1)^(k + 1) 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 + x^k)^((-1)^k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 42}]

Formula

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^((-1)^n*a(n)).

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+d+1)*d*a(d) ) * a(n-k+1).