A109083 - OEIS

%I #16 Mar 12 2021 22:24:43

%S 1,-1,0,0,-2,0,2,0,0,0,0,3,0,0,-3,0,0,0,0,0,0,-4,0,0,0,4,0,0,0,0,0,0,

%T 0,0,5,0,0,0,0,-5,0,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,6,0,0,0,0,0,0,0,0,

%U 0,0,0,0,7,0,0,0,0,0,0,-7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0

%N Convolution of A002324 and A010815.

%C Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

%H G. C. Greubel, <a href="/A109083/b109083.txt">Table of n, a(n) for n = 1..950</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F G.f.: A(x) = Sum_{n>=1} (-1)^(n-1)*n*[x^(n*(3*n-1)/2) - x^(n*(3*n+1)/2)] = x - x^2 - 2*x^5 + 2*x^7 + 3*x^12 - 3*x^15 - 4*x^22 + 4*x^26 + 5*x^35 - 5*x^40 - 6*x^51 + 6*x^57 +...

%F Expansion of f(-q)* (g(q)-1)/6 in powers of q where f(-q) = g.f. of A010815, g(q) = g.f. of A004016.

%F Expansion of (f(-q)/ f(-q^3))*(f(-q)^3 -f(-q^3) +9*q*f(-q^9)^3)/6 in powers of q where f() is a Ramanujan theta function. - _Michael Somos_, Jun 11 2006

%e G.f. satisfies 6*A(x^3) = eta(x)^3 - eta(x^3) + 3*x*eta(x^9)^3, where eta(x) = 1 + Sum_{n>=1} (-1)^n*[x^(n*(3*n-1)/2) + x^(n*(3*n+1)/2)] = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 + x^26 +... and eta(x)^3 = 1 - 3*x + 5*x^3 - 7*x^6 + 9*x^10 +... + (-1)^n*(2*n+1)*x^(n*(n+1)/2) + ... is the Jacobi triple product identity.

%t a:= CoefficientList[Series[Sum[(-1)^(n-1)*n*(x^(n*(3*n - 1)/2) - x^(n*(3*n + 1)/2)), {n,1,50}], {x,0,50}], x]; Drop[Table[a[[n]], {n, 1, 50}], 1] (* _G. C. Greubel_, May 09 2018 *)

%o (PARI) {a(n)=local(X=x+x*O(x^(3*n))); polcoeff((eta(X)^3-eta(X^3)+3*x*eta(X^9)^3)/6,3*n)}

%o (PARI) {a(n)= local(A); if(n<1, 0, A=x*O(x^n); polcoeff( eta(x+A)/ eta(x^3+A)* (eta(x+A)^3 -eta(x^3+A) +9*x* eta(x^9+A)^3)/6, n))} /* _Michael Somos_, Jun 11 2006 */

%K sign

%O 1,5

%A _Paul D. Hanna_, Jun 18 2005