|
%I #23 Jun 17 2015 12:49:25
%S 1,1,-2,5,-4,-2,8,-3,7,-4,-10,14,-12,8,8,-19,18,7,-18,20,-16,-10,24,
%T -18,21,-12,-20,40,-28,8,32,-51,20,18,-32,59,-36,-18,24,-28,42,-16,
%U -42,38,-28,24,48,-82,57,21,-36,44,-52,-20,40,-24,36,-28,-58,104,-60,32,56,-115,48,20,-66,90,-48,-32,72,-45,74,-36,-42,62,-80,24,80,-124,61,42,-82,112,-72,-42,56,-58,90,-28,-96,120,-64,48,72,-210,98,57,-70,145,-100,-36,104,-68,64,-52,-106,140,-108,40,72,-152,114,36,-96,92,-84,-58,144,-88,111
%N L.g.f.: log( 1 + Sum_{n>=1} x^(n^2) + x^(2*n^2) ).
%H Paul D. Hanna, <a href="/A256357/b256357.txt">Table of n, a(n) for n = 1..1024</a>
%H 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 L.g.f.: Sum_{n>=0} log( (1-x^(3+8*n))*(1-x^(5+8*n))*(1-x^(8+8*n)) / ( (1-x^(1+8*n))*(1-x^(4+8*n))*(1-x^(7+8*n)) ) ). [See Cooper and Hirschhorn reference]
%F From formulas given by _Michael Somos_ in A093709: (Start)
%F L.g.f.: log( (theta_3(x) + theta_3(x^2)) / 2).
%F L.g.f.: Log( psi(q^4) * f(-q^3, -q^5) / f(-q, -q^7) ) in powers of q where psi(), f() are Ramanujan theta functions.
%F L.g.f.: Log( f(-q^3, -q^5)^2 / psi(-q) ) in powers of q where psi(), f() are Ramanujan theta functions.
%F (End)
%F a(n) == 1 (mod 2) iff n is a square or twice square (A028982).
%F a(n) = -sigma(n) + [Sum_{d|n, d==2 (mod 4)} d] + [Sum_{d|n, d==1,4,7 (mod 8)} 2*d].
%e L.g.f.: L(x) = x + x^2/2 - 2*x^3/3 + 5*x^4/4 - 4*x^5/5 - 2*x^6/6 + 8*x^7/7 - 3*x^8/8 + 7*x^9/9 - 4*x^10/10 - 10*x^11/11 + 14*x^12/12 - 12*x^13/13 + 8*x^14/14 + 8*x^15/15 - 19*x^16/16 +...+ a(n)*x^n/n +...
%e where
%e exp(L(x)) = 1 + x + x^2 + x^4 + x^8 + x^9 + x^16 + x^18 + x^25 + x^32 + x^36 + x^49 + x^50 + x^64 + x^72 + x^81 + x^98 + x^100 +...+ x^A028982(n) +...
%o (PARI) {a(n) = local(L=x); L = log(1 + sum(k=1,sqrtint(n+1), x^(k^2) + x^(2*k^2)) +x*O(x^n)); n*polcoeff(L,n)}
%o for(n=1,121, print1(a(n),", "))
%o (PARI) {a(n) = -sigma(n) + sumdiv(n,d, if(d%4==2,d)) + 2*sumdiv(n,d, if((d%8)%3==1,d))}
%o for(n=1,121, print1(a(n),", "))
%o (PARI) {a(n) = local(L, X=x+x*O(x^n)); L = sum(m=0,n\8+1, log( (1-x^(3+8*m))*(1-x^(5+8*m))*(1-x^(8+8*m)) / ( (1-x^(1+8*m))*(1-x^(4+8*m))*(1-x^(7+8*m) +x*O(x^n)) ))); n*polcoeff(L,n)}
%o for(n=1,121, print1(a(n),", "))
%Y Cf. A258655, A258328, A028982, A093709.
%K sign
%O 1,3
%A _Paul D. Hanna_, Jun 03 2015
|
