A345878 G.f. A(x) satisfies: A(x) = x / exp(2 * Sum_{k>=1} A(x^k) / k).
1, -2, 5, -18, 70, -282, 1179, -5104, 22634, -102128, 467637, -2168208, 10157664, -48005858, 228607728, -1095885048, 5284044080, -25609804110, 124693451466, -609641464746, 2991742731876, -14731354000792, 72761153346680, -360397156557696, 1789733084330806, -8909067981051118
Offset: 1
Keywords
Programs
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Maple
a:= proc(n) option remember; `if`(n=1, 1, -2*add(a(n-k)*add( d*a(d), d=numtheory[divisors](k)), k=1..n-1)/(n-1)) end: seq(a(n), n=1..26); # Alois P. Heinz, Jun 28 2021 -
Mathematica
nmax = 26; A[] = 0; Do[A[x] = x/Exp[2 Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest a[1] = 1; a[n_] := a[n] = -(2/(n - 1)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 26}]
Formula
G.f.: x * Product_{n>=1} (1 - x^n)^(2*a(n)).
a(n+1) = -(2/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d) ) * a(n-k+1).