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A370541 Expansion of g.f. A(x) = Product_{n>=1} (1 + x^(n-1) + x^(2*n-1)) * (1 + x^n + x^(2*n-1)) * (1 - x^n - x^(2*n)).

%I A370541 #15 Feb 22 2024 17:36:36
%S A370541 2,5,0,-1,0,-18,-15,-23,-36,-25,-52,-35,-42,-8,13,4,96,100,208,227,
%T A370541 388,434,499,709,670,837,883,1057,775,1044,819,643,535,-78,-345,-970,
%U A370541 -1494,-3017,-3142,-5078,-6102,-7711,-9410,-11406,-13148,-15353,-17831,-18841,-22708,-22955,-26117
%N A370541 Expansion of g.f. A(x) = Product_{n>=1} (1 + x^(n-1) + x^(2*n-1)) * (1 + x^n + x^(2*n-1)) * (1 - x^n - x^(2*n)).
%C A370541 Consider function R(p,q,r) = Product_{n>=1} (1 + p^(n-1)*(q^n + r)) * (1 + p^n*(q^(n-1) + r)) * (1 - p^n*(q^n + r)) which yields Ramanujan's theta function at r = 0: R(p,q,0) = f(p,q) = Sum_{n=-oo..+oo} p^(n*(n-1)/2) * q^(n*(n+1)/2). This sequence arises from R(p,q,r) when p = x, q = x, and r = 1: A(x) = R(x,x,1).
%H A370541 Paul D. Hanna, <a href="/A370541/b370541.txt">Table of n, a(n) for n = 0..6400</a>
%e A370541 G.f.: A(x) = 2 + 5*x - x^3 - 18*x^5 - 15*x^6 - 23*x^7 - 36*x^8 - 25*x^9 - 52*x^10 - 35*x^11 - 42*x^12 - 8*x^13 + 13*x^14 + 4*x^15 + 96*x^16 + 100*x^17 + ...
%e A370541 where A(x) equals the infinite product
%e A370541 A(x) = (2 + x)*(1 + 2*x)*(1 - x - x^2) * (1 + x + x^3)*(1 + x^2 + x^3)*(1 - x^2 - x^4) * (1 + x^2 + x^5)*(1 + x^3 + x^5)*(1 - x^3 - x^6) * (1 + x^3 + x^7)*(1 + x^4 + x^7)*(1 - x^4 - x^8) * (1 + x^4 + x^9)*(1 + x^5 + x^9)*(1 - x^5 - x^10) * (1 + x^5 + x^11)*(1 + x^6 + x^11)*(1 - x^6 - x^12) * (1 + x^6 + x^13)*(1 + x^7 + x^13)*(1 - x^7 - x^14) * ...
%o A370541 (PARI) {a(n) = my(A);
%o A370541 A = prod(m=1,n+1, (1 + x^(m-1) + x^(2*m-1)) * (1 + x^m + x^(2*m-1)) * (1 - x^m - x^(2*m)) +x*O(x^n));
%o A370541 polcoeff(A,n)}
%o A370541 for(n=0,50, print1(a(n),", "))
%Y A370541 Cf. A000726, A319668.
%K A370541 sign
%O A370541 0,1
%A A370541 _Paul D. Hanna_, Feb 22 2024