cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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).

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%I A258655 #17 Jun 16 2015 22:12:48
%S A258655 1,5,7,-19,21,59,57,-115,61,145,111,-253,157,285,147,-499,307,545,343,
%T A258655 -599,399,643,553,-1501,521,889,547,-1083,813,1759,993,-2035,777,1535,
%U A258655 1197,-2359,1333,1867,1099,-3575,1723,3363,1807,-2549,1281,2765,2257,-6493,2801,3645,2149,-3503,2757
%N 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).
%C A258655 a(4*n) < 0 for n>=1, and a(n) is positive if n is not divisible by 4 (conjecture).
%H A258655 Paul D. Hanna, <a href="/A258655/b258655.txt">Table of n, a(n) for n = 1..1000</a>
%H A258655 Cooper, Shaun; Hirschhorn, Michael. <a href="http://projecteuclid.org/download/pdf_1/euclid.rmjm/1181070243">On Some Finite Product Identities.</a> Rocky Mountain J. Math. 31 (2001), no. 1, 131--139.
%F A258655 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].
%e A258655 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 +...
%e A258655 where
%e A258655 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 +...
%o A258655 (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)}
%o A258655 for(n=1, 70, print1(a(n), ", "))
%o A258655 (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))}
%o A258655 for(n=1,70, print1(a(n),", "))
%Y A258655 Cf. A258656, A256357.
%K A258655 sign
%O A258655 1,2
%A A258655 _Paul D. Hanna_, Jun 06 2015