A080918 -id:A080918 - 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

sign

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)

A072069 Number of integer solutions to the equation 2x^2+y^2+32z^2=m for an odd number m=2n-1.

Original entry on oeis.org

2, 4, 0, 0, 6, 4, 0, 0, 4, 4, 0, 0, 2, 8, 0, 0, 12, 8, 0, 0, 16, 12, 0, 0, 10, 16, 0, 0, 12, 20, 0, 0, 16, 4, 0, 0, 12, 12, 0, 0, 14, 20, 0, 0, 20, 8, 0, 0, 4, 20, 0, 0, 8, 12, 0, 0, 24, 8, 0, 0, 14, 8, 0, 0

Offset: 1

T. D. Noe, Jun 13 2002

nonn

Related to primitive congruent numbers A006991.
Assuming the Birch and Swinnerton-Dyer conjecture, the odd number 2n-1 is a congruent number if it is squarefree and 2 a(n) = A072068(n).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			a(2) = 4 because (1,1,0), (-1,1,0), (1,-1,0) and (-1,-1,0) are solutions when m=3.
G.f. = 2*x + 4*x^2 + 6*x^5 + 4*x^6 + 4*x^9 + 4*x^10 + 2*x^13 + 8*x^14 + ... - _Michael Somos_, Dec 26 2019
G.f. = 2*q + 4*q^3 + 6*q^9 + 4*q^11 + 4*q^17 + 4*q^19 + 2*q^25 + 8*q^27 + 12*q^33
+ ...

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

T. D. Noe, Table of n, a(n) for n=1..10000
Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture
Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem
Karl Rubin, Elliptic curves and right triangles

Cf. A006991, A003273, A072068, A072070, A072071, A080918.

maxN=128; soln2=Table[0, {maxN/2}]; xMax=Ceiling[Sqrt[maxN/2]]; yMax=Ceiling[Sqrt[maxN]]; zMax=Ceiling[Sqrt[maxN/32]]; Do[n=2x^2+y^2+32z^2; If[OddQ[n]&&n

{a(n) = my(A); n--; if( n<0, 0, A = x * O(x^n); polcoeff( 2 * eta(x^2 + A)^5 * eta(x^8 + A)^2 * eta(x^32 + A)^5 / (eta(x + A)^2 * eta(x^4 + A)^3 * eta(x^16 + A)^2 * eta(x^64 + A)^2), n))}; /* Michael Somos, Dec 26 2019 */

Expansion of 2 * x * phi(x) * psi(x^4) * phi(x^16) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jun 08 2012

Expansion of 2 * q^(1/2) * eta(q^2)^5 * eta(q^8)^2 * eta(q^32)^5 / (eta(q)^2 * eta(q^4)^3 * eta(q^16)^2 * eta(q^64)^2) in powers of q. - Michael Somos, Dec 26 2019

A080917 Number of integer solutions to the equation 2x^2 + y^2 + 8z^2 = n.

Original entry on oeis.org

1, 2, 2, 4, 2, 0, 4, 0, 4, 10, 4, 12, 8, 0, 8, 0, 6, 16, 6, 12, 8, 0, 4, 0, 8, 10, 12, 16, 0, 0, 8, 0, 12, 16, 8, 24, 10, 0, 12, 0, 8, 32, 8, 12, 24, 0, 8, 0, 8, 18, 14, 24, 8, 0, 16, 0, 16, 16, 4, 36, 0, 0, 16, 0, 6, 32, 16, 12, 16, 0, 8, 0, 12, 16, 20, 28, 24, 0, 8, 0, 24, 34, 8, 36, 16, 0

Offset: 0

Michael Somos, Feb 23 2003

nonn

			G.f. = 1 + 2*q + 2*q^2 + 4*q^3 + 2*q^4 + 4*q^6 + 4*q^8 + 10*q^9 + 4*q^10 + ...

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.

Cf. A000122 (theta_3(q)), A033717, A072068, A080918.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^8], {q, 0, n}]; (* Michael Somos, Feb 19 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A)^3 * eta(x^16 + A)^5 / (eta(x + A) * eta(x^8 + A)^2 * eta(x^32 + A))^2, n))};

Euler transform of period-32 sequence [2, -1, 2, -4, 2, -1, 2, 0, 2, -1, 2, -4, 2, -1, 2, -5, 2, -1, 2, -4, 2, -1, 2, 0, 2, -1, 2, -4, 2, -1, 2, -3, ...].
G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^8).
a(2*n - 1) = A072068(n). a(2*n) = A033717(n).

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

sign

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

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

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A072069 Number of integer solutions to the equation 2x^2+y^2+32z^2=m for an odd number m=2n-1.

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A080917 Number of integer solutions to the equation 2*x^2 + y^2 + 8*z^2 = n.

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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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A080917 Number of integer solutions to the equation 2x^2 + y^2 + 8z^2 = n.

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