A080964 -id:A080964 - cp's OEIS frontend

A080965 Expansion of eta(q^2)^12/(eta(q)^4eta(q^4)^5) in powers of q.

1, 4, 2, -8, -4, 8, -8, -16, 6, 12, 8, -8, -8, 24, 0, -16, 12, 16, 10, -24, -8, 16, -24, -16, 8, 28, 8, -32, -16, 8, 0, -32, 6, 32, 16, -16, -12, 40, -24, -16, 24, 16, 16, -40, -8, 40, 0, -32, 24, 36, 10, -16, -24, 24, -32, -48, 0, 32, 24, -24, -16, 40, 0, -48, 12, 16, 16

Offset: 0

Michael Somos, Feb 28 2003

sign

Euler transform of period 4 sequence [4,-8,4,-3,...].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

a(n)=A080964(4n)=2*A072071(4n)-A072070(4n).
A083703(n)=(-1)^n a(n). a(2n)=0 iff n in A004215 (checked up to n=343).
a(2n)=0 iff A005875(n)=0.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add([-3, 4, -8, 4]
      [1+irem(d, 4)]*d, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 05 2015

a[n_] := a[n] = If[n==0, 1, Sum[DivisorSum[j, {-3, 4, -8, 4}[[1 + Mod[#, 4]]]*#&]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 25 2015, after Alois P. Heinz *)

a(n)=local(X); if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X)^-4*eta(X^2)^12*eta(X^4)^-5,n))

G.f.: Product_{n>0} (1-x^(2n))^12/((1-x^n)^4(1-x^(4n))^5).

A080966 Expansion of theta_4(q^2) * theta_2(q)^2/(4*q^(1/2)) in powers of q.

Original entry on oeis.org

1, 2, -1, -2, 0, -4, -1, 2, -4, 2, 4, 2, 1, -2, 4, 2, 4, 0, -4, 0, -3, 4, -4, -4, 0, -2, 0, -6, 0, 2, -1, -4, 4, -4, -4, 8, 4, 6, 0, 2, -8, 0, 7, 2, 4, 2, 4, 0, 0, -6, 4, 0, -4, 0, 0, 0, 1, -6, -4, 4, -8, -2, -4, 4, 0, 2, -4, -6, 0, -2, 4, -8, 1, 2, 0, 0, 4, 4, 4, -2, 4, 6, 0, -2, 0, -4, -8, 10, 8, 8, -1, 4, 4, 2, -4, -4, -8, 6, 4, -6, 8, -6, 4, 4

Offset: 0

Michael Somos, Feb 28 2003

sign

The nonzero quadrisection of A248395.

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).

			q + 2*q^5 - q^9 - 2*q^13 - 4*q^21 - q^25 + 2*q^29 - 4*q^33 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Michael Somos, Introduction to Ramanujan theta functions
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

QP = QPochhammer; s = QP[q^2]^6/(QP[q]^2*QP[q^4]) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
QP := QPochhammer; a:=CoefficientList[Series[QP[q^2]^6/(QP[q]^2*QP[q^4]), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 01 2018 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^6/eta(x+A)^2/eta(x^4+A), n))}

G.f.: Product_{k>0} (1+x^k)^2*(1-x^(2k))^3/(1+x^(2k)).
Expansion of f(-q^4)*f(q)^2 in powers of q where f(-q)=f(-q,-q^2) is a Ramanujan theta function.
Expansion of q^(-1/4)*eta(q^2)^6/(eta(q)^2*eta(q^4)) in powers of q.
Euler transform of period-4 sequence [2,-4,2,-3,...].
G.f.: Product_{k>0} (1-x^(2*k))^3*(1+x^k)^2/(1+x^(2*k)).
2*a(n) = A080964(4*n+1) = 2*A072071(4*n+1) - A072070(4*n+1).

cp's OEIS Frontend

A080965 Expansion of eta(q^2)^12/(eta(q)^4eta(q^4)^5) in powers of q.

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