A317529 -id:A317529 - cp's OEIS frontend

A281542 Expansion of Sum_{i>=1} x^(i^2)/(1 + x^(i^2)) * Product_{j>=1} (1 + x^(j^2)).

1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 0, 2, 3, 0, 1, 2, 0, 0, 2, 3, 0, 0, 0, 3, 5, 0, 0, 5, 7, 0, 0, 0, 2, 3, 1, 2, 3, 4, 2, 5, 3, 0, 0, 5, 7, 0, 0, 4, 9, 4, 2, 5, 7, 5, 3, 4, 2, 3, 0, 5, 10, 4, 1, 11, 12, 0, 2, 6, 7, 4, 0, 2, 12, 12, 0, 6, 15, 9, 2, 8, 7, 3, 7, 8, 10, 9, 5, 8, 21, 13, 0, 7, 19, 13, 0, 2, 10, 13, 8

Offset: 1

Ilya Gutkovskiy, Jan 23 2017

nonn

Total number of parts in all partitions of n into distinct squares.

			a(26) = 5 because we have [25, 1], [16, 9 ,1] and 2 + 3 = 5.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index entries for related partition-counting sequences

Cf. A000290, A001422, A015723, A033461, A317529, A341040.

nmax = 100; Rest[CoefficientList[Series[Sum[x^i^2/(1 + x^i^2), {i, 1, nmax}] Product[1 + x^j^2, {j, 1, nmax}], {x, 0, nmax}], x]]

G.f.: Sum_{i>=1} x^(i^2)/(1 + x^(i^2)) * Product_{j>=1} (1 + x^(j^2)).
From Alois P. Heinz, Feb 03 2021: (Start)
a(n) = Sum_{k>=0} k * A341040(n,k).
a(n) = 0 <=> n in { A001422 }. (End)

A344299 Expansion of Sum_{k>=1} (-1)^(k+1) * x^(k^2) / (1 - x^(k^2)).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, -1, 1, 2, 1, 0, 1, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, -1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, -1, 2, 2, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 1, 2, -2, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, -1, 3, 1, 1, 0, 1, 1, 1, 0, 1, 2

Offset: 1

Ilya Gutkovskiy, May 14 2021

Number of odd squares dividing n minus number of even squares dividing n.

Inverse Moebius transform of A258998.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A046951, A048272, A072691, A258998, A317529, A344300.

nmax = 90; CoefficientList[Series[Sum[(-1)^(k + 1) x^(k^2)/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + 1) &, IntegerQ[#^(1/2)] &], {n, 1, 90}]
f[p_, e_] := 1 - (-1)^p*Floor[e/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 - (-1)^f[i,1] * floor(f[i,2]/2));} \\ Amiram Eldar, Nov 15 2022

G.f.: Sum_{k>=1} (1 - theta_4(x^k)) / 2.
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(2^e) = 1 - floor(e/2), and a(p^e) = 1 + floor(e/2) for p > 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/12 (A072691). (End)

A329801 Expansion of Sum_{k>=1} x^(k(k + 1)/2) / (1 + x^(k(k + 1)/2)).

Original entry on oeis.org

1, -1, 2, -1, 1, -1, 1, -1, 2, 0, 1, -3, 1, -1, 3, -1, 1, -1, 1, -2, 3, -1, 1, -3, 1, -1, 2, 0, 1, -1, 1, -1, 2, -1, 1, -2, 1, -1, 2, -2, 1, -2, 1, -1, 4, -1, 1, -3, 1, 0, 2, -1, 1, -1, 2, -2, 2, -1, 1, -5, 1, -1, 3, -1, 1, 0, 1, -1, 2, 0, 1, -4, 1, -1, 3, -1, 1, 0, 1, -2, 2, -1, 1, -3, 1

Offset: 1

Ilya Gutkovskiy, Nov 21 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000217, A007862, A010054, A048272, A304876, A317529.

nmax = 85; CoefficientList[Series[Sum[x^(k (k + 1)/2)/(1 + x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(-1)^(n/d + 1) Boole[IntegerQ[Sqrt[8 d + 1]]], {d, Divisors[n]}], {n, 1, 85}]

A329801(n) = sumdiv(n,d,((-1)^(1+(n/d))) * ispolygonal(d,3)); \\ Antti Karttunen, Jan 15 2025

G.f.: Sum_{k>=1} (-1)^(k + 1) * theta_2(x^(k/2)) / (2 * x^(k/8)).

a(n) = Sum_{d|n} (-1)^(n/d + 1) * A010054(d).

cp's OEIS Frontend

A281542 Expansion of Sum_{i>=1} x^(i^2)/(1 + x^(i^2)) * Product_{j>=1} (1 + x^(j^2)).

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A344299 Expansion of Sum_{k>=1} (-1)^(k+1) * x^(k^2) / (1 - x^(k^2)).

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A329801 Expansion of Sum_{k>=1} x^(k(k + 1)/2) / (1 + x^(k(k + 1)/2)).

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cp's OEIS Frontend

A281542 Expansion of Sum_{i>=1} x^(i^2)/(1 + x^(i^2)) * Product_{j>=1} (1 + x^(j^2)).

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A344299 Expansion of Sum_{k>=1} (-1)^(k+1) * x^(k^2) / (1 - x^(k^2)).

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A329801 Expansion of Sum_{k>=1} x^(k*(k + 1)/2) / (1 + x^(k*(k + 1)/2)).

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A329801 Expansion of Sum_{k>=1} x^(k(k + 1)/2) / (1 + x^(k(k + 1)/2)).