A258655 - OEIS

A258655

a(n) = A256357(n^2), where exp( Sum_{n>=1} A256357(n)*x^n/n ) = 1 + Sum_{n>=1} x^(n^2) + x^(2*n^2).

1, 5, 7, -19, 21, 59, 57, -115, 61, 145, 111, -253, 157, 285, 147, -499, 307, 545, 343, -599, 399, 643, 553, -1501, 521, 889, 547, -1083, 813, 1759, 993, -2035, 777, 1535, 1197, -2359, 1333, 1867, 1099, -3575, 1723, 3363, 1807, -2549, 1281, 2765, 2257, -6493, 2801, 3645, 2149, -3503, 2757

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OFFSET

1,2

COMMENTS

a(4*n) < 0 for n>=1, and a(n) is positive if n is not divisible by 4 (conjecture).

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cooper, Shaun; Hirschhorn, Michael. On Some Finite Product Identities. Rocky Mountain J. Math. 31 (2001), no. 1, 131--139.

FORMULA

a(n) = -sigma(n^2) + [Sum_{d|n^2, d==2 (mod 4)} d] + [Sum_{d|n^2, d==1,4,7 (mod 8)} 2*d].

EXAMPLE

L.g.f.: L(x) = x + 5*x^2/2 + 7*x^3/3 - 19*x^4/4 + 21*x^5/5 + 59*x^6/6 + 57*x^7/7 - 115*x^8/8 + 61*x^9/9 + 145*x^10/10 +...+ A256357(n^2)*x^n/n +...

where

exp(L(x)) = 1 + x + 3*x^2 + 5*x^3 + 2*x^4 + 10*x^5 + 13*x^6 + 23*x^7 + 43*x^8 + 57*x^9 + 66*x^10 +...+ A258656(n)*x^n +...

PROG

(PARI) {a(n) = local(L=x); L = log(1 + sum(k=1, n+1, x^(k^2) + x^(2*k^2)) +x*O(x^(n^2))); n^2*polcoeff(L, n^2)}

for(n=1, 70, print1(a(n), ", "))

(PARI) {a(n) = -sigma(n^2) + sumdiv(n^2, d, if(d%4==2, d)) + 2*sumdiv(n^2, d, if((d%8)%3==1, d))}

for(n=1, 70, print1(a(n), ", "))

CROSSREFS

Cf. A258656, A256357.

Sequence in context: A046151 A046078 A075409 * A174362 A268608 A058079

Adjacent sequences: A258652 A258653 A258654 * A258656 A258657 A258658

KEYWORD

sign

AUTHOR

Paul D. Hanna, Jun 06 2015

STATUS

approved