A363087 - OEIS

%I #7 May 19 2023 08:03:51

%S 1,-1,-1,1,2,-1,-5,-1,11,10,-21,-39,30,126,4,-354,-261,834,1347,-1483,

%T -5033,823,15663,8765,-41112,-56364,84888,234546,-91319,-791833,

%U -293380,2251507,2561264,-5177875,-11835968,7620048,42944358,7464956,-130615874,-119900209

%N 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 + ...).

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

%F 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).

%t 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

%t 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}]

%Y Cf. A007560, A049075, A345234, A363062.	

%K sign

%O 1,5

%A _Ilya Gutkovskiy_, May 18 2023