A034950 -id:A034950 - cp's OEIS frontend

A248394 q-Expansion of the modular form of weight 3/2, g*theta(2) in Tunnell's notation (see Comments).

0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, -2, 0, 0, 0, 0, 0, -4, 0, -2, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 4, 0, -4, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 4, 0, -2

Offset: 0

N. J. A. Sloane, Oct 18 2014

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g = q*Product_{m=1..oo} (1-q^(8*m))*(1-q^(16*m)),
theta(t) = Sum_{n=-oo..oo} q^(t*n^2).
Although the OEIS does not normally include sequences in which every other term is zero, this one is important enough to warrant an exception.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
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, Jacobi Theta Functions

The nonzero bisection is A034950, which has further information and references.
Used in A248397-A248406.
Cf. A000122 (theta_3(q)), A072068, A072069, A080917, A080918, A248395.

# This produces a list of the first 100 terms:
g:=q*mul((1-q^(8*m))*(1-q^(16*m)),m=1..30);
g:=series(g,q,100);
th:=t->series( add(q^(t*n^2),n=-50..50), q, 100);
series(g*th(2),q,100);
seriestolist(%);
# Alternative with https://oeis.org/transforms.txt and the Somos Euler transform in A034950:
p8 := [2,-3,2,-2,2,-3,2,-3] ;
L := [seq(op(p8),i=1..10)] ;
EULER(%) ;
[1,op(%)] ;
[0,op(AERATE(%,1))] ; # R. J. Mathar, Nov 11 2014

QP = QPochhammer; s = q*QP[q^8]*QP[q^16]*EllipticTheta[3, 0, q^2] + O[q]^80; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

From Seiichi Manyama, Sep 30 2018: (Start)
Let q = exp(Pi i t).
theta_3(q) = 1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + ... .
G.f.: (theta_3(q) - theta_3(q^4))*(theta_3(q^32) - theta_3(q^8)/2)*theta_3(q^2).
a(2*n-1) = A080918(2*n-1) - A080917(2*n-1)/2 = A072069(n) - A072068(n)/2 for n > 0. (End)

A080963 Expansion of theta_3(q)theta_3(q^2)theta_4(q^8) in powers of q.

Original entry on oeis.org

1, 2, 2, 4, 2, 0, 4, 0, 0, 2, -4, -4, 0, 0, -8, 0, -2, -8, 6, -4, -8, 0, 4, 0, 0, -6, -12, 0, 0, 0, -8, 0, -4, 8, 8, -8, 10, 0, 12, 0, 0, 0, -8, 12, 0, 0, -8, 0, 8, 2, 14, 8, -8, 0, 16, 0, 0, 8, -4, 4, 0, 0, -16, 0, 6, 0, 16, -4, 16, 0, 8, 0, 0, 8, -20, -4, 0, 0, -8, 0, -8, -6, 8, 4, -16, 0, 20, 0, 0, -8, -20, -8, 0, 0, -16, 0, -8

Offset: 0

Michael Somos, Feb 28 2003

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Ramanujan theta functions: f(q) := Product_{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) := Product_{k>=0} (1+q^(2k+1)) (A000700).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

a[n_]:= SeriesCoefficient[EllipticTheta[3,0,q]*EllipticTheta[3,0,q^2]* EllipticTheta[3,0,-q^8], {q,0,n}]; Table[a[n], {n,0,50}] (* or *)
eta[q_] := q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[ (eta[q^2]*eta[q^4])^3/(eta[q]^2*eta[q^16]), {q,0,n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Feb 11 2018 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^4+A))^3/(eta(x+A)^2*eta(x^16+A)), n))}

a(16*n+5) = a(16*n+7) = a(16*n+8) = a(16*n+12) = a(16*n+13) = a(16*n+15) = 0.
a(n) = 2*A080918(n) - A080917(n).
a(2*n+1) = 2*A034950(n).
Expansion of (eta(q^2)*eta(q^4))^3/(eta(q)^2*eta(q^16)) in powers of q.
Euler transform of period-16 sequence [2,-1,2,-4,2,-1,2,-4,2,-1,2,-4,2,-1,2,-3,...].
Expansion of phi(q)phi(q^2)phi(-q^8) in powers of q where phi() is a Ramanujan theta function.
G.f.: Product_{k>0} (1+x^k)^2*(1-x^(2k))*(1-x^(4k))^2/((1+x^(4k))*(1+x^(8k))). - Michael Somos, Feb 16 2006

A255252 Expansion of psi(x) * psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, -1, 0, -2, 3, 2, 1, -1, -1, 1, -2, 1, -3, -2, -2, 3, 1, -1, 4, 3, -1, -1, 2, -4, 4, 1, 0, -1, -2, -3, -3, -4, 2, 3, -3, 0, 0, 5, 2, 0, -3, 2, -1, 4, 1, 0, 1, 3, 0, -2, 2, -1, -2, -4, -5, 2, 0, -7, 3, -4, 3, 1, 5, 2, -5, -1, -1, -3, 4, -1, 3, 4, 1, 4

Offset: 0

Michael Somos, Feb 18 2015

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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - x - x^2 - 2*x^4 + 3*x^5 + 2*x^6 + x^7 - x^8 - x^9 + x^10 + ...
G.f. = q^3 - q^11 - q^19 - 2*q^35 + 3*q^43 + 2*q^51 + q^59 - q^67 - q^75 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A034950.

A255252 := proc(n)
    local psi,x,i ;
    psi := add( A010054(i)*x^i,i=0..n) ;
    psi*subs(x=-x,psi)^2 ;
    coeftayl(%,x=0,n) ;
end proc:
seq(A255252(n),n=0..20) ; # R. J. Mathar, Feb 22 2021

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^4]^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(1/2)]^2 / (4 x^(3/8)), {x, 0, n}];

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

Expansion of f(-x) * f(-x^4)^2 in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-3/8) * eta(q) * eta(q^4)^2 in powers of q.
Euler transform of period 4 sequence [ -1, -1, -1, -3, ...].
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(4*k))^2.
2 * a(n) = A034950(4*n + 1).

A255257 Expansion of psi(x) * phi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, -4, -3, 4, 0, 1, 4, 0, 4, -3, -4, -4, -8, 8, 1, -4, 0, 0, 4, 0, 5, 4, 8, -4, -4, 4, -8, -3, -4, 4, -4, 0, 0, -8, 4, 1, 0, -8, 0, 4, 8, 8, 8, 0, 1, 0, -8, 8, -4, -4, -8, 12, 4, -12, 1, -4, 0, 0, -4, -8, 4, -8, 0, 0, -8, 1, 12, 8, 8, 0, -8, 8, 0, 8, 4, 0

Offset: 0

Michael Somos, Feb 19 2015

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x - 4*x^2 - 3*x^3 + 4*x^4 + x^6 + 4*x^7 + 4*x^9 - 3*x^10 + ...
G.f. = q + q^9 - 4*q^17 - 3*q^25 + 4*q^33 + q^49 + 4*q^57 + 4*q^73 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A034950.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2] ^2 EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}];

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

Expansion of q^(-1/8) * eta(q^2)^6 / (eta(q) * eta(q4)^2) in powers of q.
Euler transform of period 4 sequence [ 1, -5, 1, -3, ...].
a(n) = A034950(4*n).

cp's OEIS Frontend

A248394 q-Expansion of the modular form of weight 3/2, g*theta(2) in Tunnell's notation (see Comments).

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A080963 Expansion of theta_3(q)*theta_3(q^2)*theta_4(q^8) in powers of q.

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A255252 Expansion of psi(x) * psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.

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Maple

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A255257 Expansion of psi(x) * phi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

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Mathematica

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A080963 Expansion of theta_3(q)theta_3(q^2)theta_4(q^8) in powers of q.