A304876 L.g.f.: log(Product_{k>=1} (1 + x^(k*(k+1)/2))) = Sum_{n>=1} a(n)*x^n/n.

1, -1, 4, -1, 1, 2, 1, -1, 4, 9, 1, -10, 1, -1, 19, -1, 1, 2, 1, -11, 25, -1, 1, -10, 1, -1, 4, 27, 1, -3, 1, -1, 4, -1, 1, 26, 1, -1, 4, -11, 1, -19, 1, -1, 64, -1, 1, -10, 1, 9, 4, -1, 1, 2, 56, -29, 4, -1, 1, -35, 1, -1, 25, -1, 1, 68, 1, -1, 4, 9, 1, -46, 1, -1, 19, -1, 1

Offset: 1

Author

Ilya Gutkovskiy, May 20 2018

Keywords

• sign

Examples

			L.g.f.: L(x) = x - x^2/2 + 4*x^3/3 - x^4/4 + x^5/5 + 2*x^6/6 + x^7/7 - x^8/8 + 4*x^9/9 + 9*x^10/10 + ...
exp(L(x)) = 1 + x + x^3 + x^4 + x^6 + x^7 + x^9 + 2*x^10 + x^11 + x^13 + x^14 + x^15 + 2*x^16 + ... + A024940(n)*x^n + ...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences related to sums of divisors

Crossrefs

Cf. A000217, A010054, A024940, A185027, A300853.

Programs

Mathematica

nmax = 77; Rest[CoefficientList[Series[Log[Product[1 + x^(k (k + 1)/2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 77; Rest[CoefficientList[Series[Sum[k (k + 1)/2 x^(k (k + 1)/2)/(1 + x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, IntegerQ[(8 # + 1)^(1/2)] &], {n, 77}]

PARI

A010054(n) = issquare(8*n + 1);
A304876(n) = sumdiv(n,d,(-1)^(1+(n/d)) * A010054(d)*d); \\ Antti Karttunen, Feb 20 2023

Formula

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

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