A307992 - OEIS

%I #9 May 09 2019 16:13:05

%S 1,1,1,3,4,9,9,20,16,38,28,61,39,110,52,149,84,225,101,317,120,454,

%T 175,543,198,823,243,940,327,1259,356,1601,387,2051,515,2270,623,3114,

%U 660,3373,829,4381,870,5145,913,6264,1245,6683,1292,8776,1404,9477,1724

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

%H Alois P. Heinz, <a href="/A307992/b307992.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%p a:= proc(n) option remember; `if`(n<3, signum(n), (m->

%p       m*add(a(d)/d, d=numtheory[divisors](m)))(n-2))

%p     end:

%p seq(a(n), n=1..60);  # _Alois P. Heinz_, May 09 2019

%t 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]]

%t 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}]

%t 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}]

%Y Cf. A007439, A038046, A144366, A318372.

%K nonn

%O 1,4

%A _Ilya Gutkovskiy_, May 09 2019