A089807 -id:A089807 - cp's OEIS frontend

A089801 a(n) = 0 unless n = 3j^2 + 2j or 3j^2 + 4j + 1 for some j >= 0, in which case a(n) = 1.

1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 0

Eric W. Weisstein, Nov 12 2003

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also characteristic function of generalized octagonal numbers A001082. - Omar E. Pol, Jul 13 2012
Number 12 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

			G.f. = 1 + x + x^5 + x^8 + x^16 + x^21 + x^33 + x^40 + x^56 + x^65 + x^85 + ...
G.f. = q + q^4 + q^16 + q^25 + q^49 + q^64 + q^100 + q^121 + q^169 + q^196 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See Th. 2.
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See X(q).
James A. Sellers, Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice, arXiv:2409.12321 [math.NT], 2024. See p. 4.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990.
Index entries for characteristic functions.

Cf. A001082, A033684, A080995, A089802, A089807, A280739.

Basis( ModularForms( Gamma0(36), 1/2), 87) [2]; /* Michael Somos, Jul 02 2014 */

A089801 := proc(n)
        A033684(3*n+1) ;
end proc: # R. J. Mathar, Oct 07 2011
M:=33;
S:=f->series(f,q,500);
L:=f->seriestolist(f);
X:=add(q^(3*n^2+2*n),n=-M..M);
S(%);
L(%); # N. J. A. Sloane, Jan 31 2012
eps:=Array(0..120,0);
for j from 0 to 120 do
if 3*j^2+2*j <= 120 then eps[3*j^2+2*j] := 1; fi;
if 3*j^2+4*j+1 <= 120 then eps[3*j^2+4*j+1] := 1; fi;
end do;  # N. J. A. Sloane, Aug 12 2017

a[ n_] := SeriesCoefficient[ (1/2) x^(-1/3) (EllipticTheta[ 3, 0, x^(1/3)] - EllipticTheta[ 3, 0, x^3]), {x, 0, n}]; (* Michael Somos, Jun 29 2012 *)
a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/8) QPochhammer[ -x, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}]; (* Michael Somos, Jun 29 2012 *)

{a(n) = issquare(3*n + 1)}; /* Michael Somos, Apr 12 2005 */

from sympy.ntheory.primetest import is_square
def A089801(n): return int(is_square(3*n+1)) # Chai Wah Wu, May 17 2023

G.f.: Sum_{n=-oo..oo} q^(3n^2+2n).
Expansion of Jacobi theta function (theta_3(q^(1/3)) - theta_3(q^3))/(2 q^(1/3)) in powers of q.
Euler transform of period 12 sequence [1, -1, 0, 0, 1, -1, 1, 0, 0, -1, 1, -1, ...]. - Michael Somos, Apr 12 2005
a(n) = b(3*n + 1) where b() is multiplicative with b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p != 3. - Michael Somos, Jun 06 2005; b=A033684. - R. J. Mathar, Oct 07 2011
Expansion of q^(-1/3) * eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)) in powers of q. - Michael Somos, Apr 12 2005
Expansion of chi(x) * psi(-x^3) in powers of x where chi(), psi() are Ramanujan theta functions. - Michael Somos, Apr 19 2007
Expansion of f(x, x^5) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Jun 29 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089807.
a(8*n + 4) = a(4*n + 2) = a(4*n + 3) = 0, a(4*n + 1) = a(n), a(8*n) = A080995(n). - Michael Somos, Jan 28 2011
a(n) = (-1)^n * A089802(n).
For n > 0, a(n) = b(n)-b(n-1) + c(n)-c(n-1), where b(n) = floor(sqrt(n/3+1/9)+2/3) and c(n) = floor(sqrt(n/3+1/9)+4/3). - Mikael Aaltonen, Jan 22 2015
a(n) = A033684(3*n + 1). - Michael Somos, Jan 10 2017

Edited with simpler definition by N. J. A. Sloane, Jan 31 2012

Further edited by N. J. A. Sloane, Aug 12 2017

A089810 Expansion of Jacobi theta function theta_3(Pi/6, q) in powers of q.

Original entry on oeis.org

1, 1, 0, 0, -1, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0

Offset: 0

Eric W. Weisstein, Nov 12 2003

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = -x^5, b = -x. - Michael Somos, Jul 12 2012
Convolution square is A258279. - Michael Somos, May 25 2015
Number 8 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

			G.f. = 1 + q - q^4 - 2*q^9 - q^16 + q^25 + 2*q^36 + q^49 - q^64 - 2*q^81 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Eric Weisstein's World of Mathematics, Quintuple Product Identity.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990.

Cf. A002448, A080995, A089801, A089802, A089807, A089812, A204843, A204853, A258279.

Cf. A000700, A000122, A010054, A121373.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, Pi/6, q], {q, 0, n}]; (* Michael Somos, Nov 14 2011 *)
a[ n_] := SeriesCoefficient[ (3 EllipticTheta[ 4, 0, q^9] - EllipticTheta[ 4, 0, q])  /2, {q, 0, n}]; (* Michael Somos, Nov 14 2011 *)
QP = QPochhammer; s = QP[q^2]^2*(QP[q^3] / (QP[q]*QP[q^6])) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015, adapted from PARI *)

{a(n) = my(x); if( n<1, n==0, issquare(n, &x) * (1 + (n%3==0)) * (-1)^((1 + x) \ 3))}; /* Michael Somos, Nov 05 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)), n))}; /* Michael Somos, Jan 26 2008 */

Expansion of Jacobi theta function (3theta_4(q^9) - theta_4(q)) / 2 in powers of q.
a(n) is multiplicative with  a(0)=1, a(2^e) = -(1 + (-1)^e)/2, if e>0, a(3^e) = -2(1 + (-1)^e)/2 if e>0, a(p^e) = (1 + (-1)^e)/2 otherwise.
From Michael Somos, Nov 05 2005: (Start)
Euler transform of period 6 sequence [ 1, -1, 0, -1, 1, -1, ...].
G.f.: (Sum_{k in Z} 3 * (-x)^((3*k)^2) - (-x)^(k^2)) / 2 = Product_{k>0} (1 - x^(2*k)) / ((1 - x^(6*k - 1)) * (1 - x^(6*k-5))).
Expansion of eta(q^2)^2 * eta(q^3) / (eta(q) * eta(q^6)) in powers of q. (End)
Expansion of psi(q) * chi(-q^3) in powers of q where psi(), chi() are Ramanujan theta functions. - Michael Somos, Sep 16 2007
Expansion of (3 * phi(-q^9) - phi(-q)) / 2 in powers of q where phi() is a Ramanujan theta function.
From Michael Somos, Sep 17 2007: (Start)
Expansion of Jacobi theta function theta_3(Pi/6, q) in powers of q.
Expansion of f(x*w, x/w) in powers of x where w is a primitive sixth root of unity and f() is Ramanujan's two-variable theta function. (End)
From Michael Somos, Jan 26 2008: (Start)
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 72^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A080995.
G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^k + x^(2*k)). (End)
a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 3) = a(8*n + 5) = a(9*n + 3) = a(9*n + 6) = 0. a(3*n + 1) = A089802(n). a(4*n) = A089807(n). a(9*n) = A002448(n).
a(n) = (floor(sqrt(n))-floor(sqrt(n-1)))*(abs(2-4*sin((floor(sqrt(n))+1)*Pi/3)^2) - 4*sin((floor(sqrt(n))+2)*Pi/3)^2)*(-1)^floor(floor(sqrt(n)-1)/3). - Mikael Aaltonen, Jan 17 2015
From Michael Somos, May 25 2015: (Start)
a(n) = (-1)^n * A089807(n) = A204843(4*n) = A204853(4*n).
a(8*n + 1) = A089812(n). a(12*n + 4) = - A089801(n). (End)
Sum_{k=1..n} abs(a(k)) ~ (4/3)*sqrt(n). - Amiram Eldar, Jan 27 2024

A258292 Expansion of psi(-q)^2 * chi(q^3)^2 in powers of q where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, May 25 2015

sign

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

			G.f. = 1 - 2*q + q^2 - 2*q^4 + 2*q^5 + q^8 + 4*q^9 - 4*q^10 - 4*q^13 + ...kkj

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. A002175, A004018, A089807, A122856, A122865, A258228, A258279.

A := Basis( ModularForms( Gamma1(36), 1), 82); A[1] - 2*A[2] + A[3] - 2*A[5] + 2*A[6] + A[9] + 4*A[10] - 4*A[11] - 4*A[14] - 2*A[17] + 2*A[18
] + 4*A[19];

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

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)))^2, n))};

Expansion of f(q) * psi(-q)^2 / psi(-q^3) in powers of q where psi(), f() are Ramanujan theta functions.
Expansion of f(x*w, x/w)^2 in powers of x where w is a primitive cube root of unity and f() is Ramanujan's general theta function.
Expansion of (eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^12)))^2 in powers of q.
Euler transform of period 12 sequence [ -2, 0, 0, -2, -2, -2, -2, -2, 0, 0, -2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 18 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A122856.
G.f.: (Product_{k>0} (1 + x^k) * (1 + x^(3*k)) / (1 - x^(2*k) + x^(4*k)))^2.
a(n) = (-1)^n * A258279(n).  Convolution square of A089807.
a(2*n) = A258228(n). a(3*n + 1) = -2 * A122865(n). a(3*n + 2) = A122856(n). a(4*n) = a(n). a(4*n + 3) = 0. a(12*n + 1) = -2 * A002175(n).
a(18*n) = A004018(n). a(18*n + 3) = a(18*n + 6) = a(18*n + 12) = 0.

A245668 Expansion of (chi(q^3) * psi(-q))^3 in powers of q where chi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -3, 3, -1, -3, 6, -3, 0, 3, 3, -12, 6, -1, -12, 12, 0, -3, 12, 9, -12, 6, -6, -12, 0, -3, -15, 18, 5, 0, 18, -6, 0, 3, -6, -24, 12, 3, -12, 18, 0, -12, 24, -6, -12, 6, 18, -24, 0, -1, -27, 21, -6, -12, 18, 15, 0, 12, -6, -12, 18, 0, -36, 24, 0, -3, 24, -12

Offset: 0

Michael Somos, Jul 28 2014

sign

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

			G.f. = 1 - 3*q + 3*q^2 - q^3 - 3*q^4 + 6*q^5 - 3*q^6 + 3*q^8 + 3*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A089807, A213592, A213607, A245669.

A := Basis( ModularForms( Gamma0(12), 3/2), 67);  A[1] - 3*A[2] + 3*A[3];

a[ n_] := SeriesCoefficient[ EllipticTheta[3, Pi/3, q]^3, {q,0,n}];
a[ n_] := SeriesCoefficient[ ((3 EllipticTheta[3, 0, q^9] - EllipticTheta[3, 0, q]) / 2)^3, {q,0,n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[-q^3, q^6] EllipticTheta[2, 0, Sqrt[-q]] / (2 (-q)^(1/8)))^3, {q,0,n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)))^3, n))};

Expansion of phi(q^3) * psi(-q)^3 / psi(-q^3) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^12)))^3 in powers of q.
Euler transform of period 12 sequence [-3, 0, 0, -3, -3, -3, -3, -3, 0, 0, -3, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A245669.
a(3*n + 1) = -3 * A213056(n). a(6*n + 2) = 3 * A213592(n). a(6*n + 5) = 6 * A213607(n). a(8*n + 7) = 0.
Convolution cube of A089807.

cp's OEIS Frontend

A089801 a(n) = 0 unless n = 3j^2 + 2j or 3j^2 + 4j + 1 for some j >= 0, in which case a(n) = 1.

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A089810 Expansion of Jacobi theta function theta_3(Pi/6, q) in powers of q.

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A258292 Expansion of psi(-q)^2 * chi(q^3)^2 in powers of q where psi(), f() are Ramanujan theta functions.

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A245668 Expansion of (chi(q^3) * psi(-q))^3 in powers of q where chi(), psi() are Ramanujan theta functions.

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