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Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(i*j*(j + 1)/2)).
0

%I #5 Sep 24 2019 22:00:56

%S 1,1,2,5,10,21,47,99,211,455,973,2081,4464,9558,20466,43848,93914,

%T 201140,430844,922818,1976553,4233613,9067960,19422576,41601229,

%U 89105550,190854784,408791400,875589076,1875421302,4016959325,8603912899,18428694036,39472363286

%N Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(i*j*(j + 1)/2)).

%C Invert transform of A007862.

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

%F a(0) = 1; a(n) = Sum_{k=1..n} A007862(k) * a(n-k).

%t nmax = 33; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1)/2)/(1 - x^(k (k + 1)/2)), {k, 1, nmax}]), {x, 0, nmax}], x]

%t a[0] = 1; a[n_] := a[n] = Sum[Length[Select[Divisors[k], IntegerQ[Sqrt[8 # + 1]] &]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

%Y Cf. A007862, A129921, A327738, A327744, A327745.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Sep 24 2019