A307992 G.f. A(x) satisfies: A(x) = x + x^2 * (1 + A(x) + 2*A(x^2) + 3*A(x^3) + ...).

1, 1, 1, 3, 4, 9, 9, 20, 16, 38, 28, 61, 39, 110, 52, 149, 84, 225, 101, 317, 120, 454, 175, 543, 198, 823, 243, 940, 327, 1259, 356, 1601, 387, 2051, 515, 2270, 623, 3114, 660, 3373, 829, 4381, 870, 5145, 913, 6264, 1245, 6683, 1292, 8776, 1404, 9477, 1724

Offset

1,4

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

G.f.: x + x^2 * (1 + Sum_{n>=1} a(n)*x^n/(1 - x^n)^2).

a(1) = a(2) = 1; a(n+2) = Sum_{d|n} d*a(n/d).

Maple

a:= proc(n) option remember; `if`(n<3, signum(n), (m->
      m*add(a(d)/d, d=numtheory[divisors](m)))(n-2))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, May 09 2019

Mathematica

terms = 57; A[_] = 0; Do[A[x_] = x + x^2 (1 + Sum[k A[x^k], {k, 1, terms}]) + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
a[n_] := a[n] = SeriesCoefficient[x + x^2 (1 + Sum[a[k] x^k/(1 - x^k)^2, {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 57}]
a[n_] := a[n] = Sum[d a[(n - 2)/d], {d, Divisors[n - 2]}]; a[1] = a[2] = 1; Table[a[n], {n, 1, 57}]

Crossrefs

Cf. A007439, A038046, A144366, A318372.

Sequence in context: A202499 A341319 A063780 * A323180 A095047 A246261

Adjacent sequences: A307989 A307990 A307991 * A307993 A307994 A307995

Keyword

nonn

Author

Ilya Gutkovskiy, May 09 2019

Status

approved