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G.f.: exp( Sum_{n>=1} A002129(n^2)*x^n/n ), where exp( Sum_{n>=1} A002129(n)*x^n/n ) = Sum_{n>=0} x^() is the excess of sum of odd divisors of n*( over sum of even divisors of n+1)/2)..

A002129(n) is the excess of sum of odd divisors of n over sum of even divisors of n.

Compare to: Sum_{n>=0} x^(n*(n+1)/2) = exp( Sum_{n>=1} A002129(n)*x^n/n ).

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Cf. A224340, A224339, A002129; variant: A215603.

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G.f.: exp( Sum_{n>=1} A002129(n^2)*x^n/n ), where exp( Sum_{n>=1} A002129(n)*x^n/n ) = Sum_{n>=0} x^(n*(n+1)/2).

A002129(n) is the excess of sum of odd divisors of n over sum of even divisors of n.

Paul D. Hanna, <a href="/A225925/b225925.txt">Table of n, a(n) for n = 0..1000</a>

G.f.: A(x) = 1 + x - 2*x^2 + 2*x^3 - x^4 - 7*x^5 + 8*x^6 - 14*x^7 + x^8 +...

((PARI) {A002129(n)=if(n<1, 0, -sumdiv(n, d, (-1)^d*d))}

Cf. A224340, A224339, A002129.

allocated for Paul D. Hanna

G.f.: exp( Sum_{n>=1} A002129(n^2)*x^n/n ), where exp( Sum_{n>=1} A002129(n)*x^n/n ) = Sum_{n>=0} x^(n*(n+1)/2).

1, 1, -2, 2, -1, -7, 8, -14, 1, 11, -23, 43, -54, 38, 17, -55, 162, -198, 257, -175, 69, 141, -518, 764, -1049, 1215, -1241, 549, 161, -1625, 3192, -5176, 6782, -7568, 7267, -4263, -788, 8394, -17866, 29782, -39041, 46101, -45857, 36551, -14591, -20937, 70638, -129520, 190994, -245846, 280560

0,3

A002129(n) is the excess of sum of odd divisors of n over sum of even divisors of n.

G.f.: A(x) = 1 + x - 2*x^2 + 2*x^3 - x^4 - 7*x^5 + 8*x^6 - 14*x^7 + x^8 +...

where

log(A(x)) = x - 5*x^2/2 + 13*x^3/3 - 29*x^4/4 + 31*x^5/5 - 65*x^6/6 + 57*x^7/7 - 125*x^8/8 + 121*x^9/9 - 155*x^10/10 +...+ A002129(n^2)*x^n/n +...

(PARI) {A002129(n)=if(n<1, 0, -sumdiv(n, d, (-1)^d*d))}

{a(n)=polcoeff(exp(sum(k=1, n, A002129(k^2)*x^k/k)+x*O(x^n)), n)}

for(n=0, 50, print1(a(n), ", "))

Cf. A224340, A224339, A002129.

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Paul D. Hanna, May 20 2013

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