A345884 G.f. A(x) satisfies: A(x) = x * exp(2 * Sum_{k>=1} (-1)^k * A(x^k) / k).

1, -2, 7, -26, 103, -442, 1982, -9122, 42985, -206526, 1007322, -4974066, 24819268, -124949782, 633882799, -3237261340, 16629986395, -85873762466, 445491479309, -2320717519612, 12134813554225, -63667883444468, 335083404759136, -1768545061282712, 9358571746569760

Offset

1,2

Links

Table of n, a(n) for n=1..25.

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} (-1)^(k/d) * d * a(d) ) * a(n-k+1).

Maple

a:= proc(n) option remember; `if`(n=1, 1, 2*add(a(n-k)*add(d*a(d)
       *(-1)^(k/d), d=numtheory[divisors](k)), k=1..n-1)/(n-1))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 28 2021

Mathematica

nmax = 25; A[_] = 0; Do[A[x_] = x Exp[2 Sum[(-1)^k 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[(-1)^(k/d) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 25}]

Crossrefs

Cf. A000151, A005753, A049075, A345885.

Sequence in context: A150542 A150543 A150544 * A150545 A150546 A151296

Adjacent sequences: A345881 A345882 A345883 * A345885 A345886 A345887

Keyword

sign

Author

Ilya Gutkovskiy, Jun 28 2021

Status

approved