A210825 - OEIS

%I #13 Apr 05 2012 21:59:18

%S 1,0,-1,0,-1,0,0,-1,0,0,0,0,-1,-1,1,1,-1,0,-1,0,0,0,-1,1,0,0,0,0,0,0,

%T -1,-1,0,0,0,0,0,0,1,1,-1,1,-1,-1,0,1,-1,-1,-1,0,1,0,-1,0,0,2,1,-1,-1,

%U 0,-1,0,0,0,1,0,0,0,1,1,-1,0,-1,-1,0,0,-1,0,0,-1,0,0,-1

%N G.f.: Sum_{n>=1} a(n)*x^n/(1-x^n)  =  x*Sum_{n>=1} x^(n*(n-1)/2).

%H Paul D. Hanna, <a href="/A210825/b210825.txt">Table of n, a(n) for n = 1..1035</a>

%e G.f.: x/(1-x) - x^3/(1-x^3) - x^5/(1-x^5) - x^8/(1-x^8) - x^13/(1-x^13) - x^14/(1-x^14) + x^15/(1-x^15) + x^16/(1-x^16) - x^17/(1-x^17) +...+ a(n)*x^n/(1-x^n) +...

%e = x + x^2 + x^4 + x^7 + x^11 + x^16 + x^22 + x^29 +...+ x^(n*(n-1)/2+1) +...

%o (PARI) {a(n)=local(TRI=sum(k=1,sqrtint(4*n+1),x^(k*(k-1)/2)));if(n==1,1,polcoeff(x*TRI-sum(m=1,n-1,a(m)*x^m/(1-x^m+x*O(x^n))),n))}

%o for(n=1,30,print1(a(n),", "))

%o (PARI) /* Vectorized form (faster): */

%o {A=[1];for(i=1,256,print1(A[#A],", ");A=concat(A,0);A[#A]=polcoeff(x*sum(k=1,sqrtint(2*#A)+1,x^(k*(k-1)/2)) - sum(m=1,#A-1,A[m]*x^m/(1-x^m+x*O(x^#A))),#A));print1(A[#A])}

%o {sum(n=1,#A,A[n]*x^n/(1-x^n+O(x^(#A))))} /* Verify Lambert series */

%Y Cf. A008836.

%K sign

%O 1,56

%A _Paul D. Hanna_, Mar 27 2012