A327745 Expansion of Product_{i>=1, j>=1} (1 + x^(i*j*(j + 1)/2)).

1, 1, 1, 3, 3, 4, 8, 9, 11, 19, 23, 28, 42, 51, 62, 89, 108, 130, 178, 215, 260, 344, 413, 496, 639, 766, 916, 1155, 1380, 1641, 2040, 2426, 2870, 3520, 4166, 4912, 5960, 7023, 8246, 9911, 11634, 13610, 16224, 18972, 22111, 26183, 30507, 35430, 41698

Offset

0,4

Comments

Weigh transform of A007862.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Formula

G.f.: Product_{k>=1} (1 + x^k)^A007862(k).

Mathematica

nmax = 48; CoefficientList[Series[Product[(1 + x^k)^Length[Select[Divisors[k], IntegerQ[Sqrt[8 # + 1]] &]], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d Length[Select[Divisors[d], IntegerQ[Sqrt[8 # + 1]] &]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]
nmax = 50; CoefficientList[Series[Product[QPochhammer[-1, x^(k*(k + 1)/2)]/2, {k, 1, Sqrt[2*nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 24 2019 *)

Crossrefs

Cf. A007862, A280451, A327744.

Sequence in context: A137418 A357306 A285445 * A155822 A160646 A019466

Adjacent sequences: A327742 A327743 A327744 * A327746 A327747 A327748

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 23 2019

Status

approved