A258279 -id:A258279 - cp's OEIS frontend

A002175 Excess of number of divisors of 12n+1 of form 4k+1 over those of form 4k+3.

1, 2, 3, 2, 1, 2, 2, 4, 2, 2, 1, 0, 4, 2, 3, 2, 2, 4, 0, 2, 2, 0, 4, 2, 3, 0, 2, 6, 2, 2, 1, 2, 0, 2, 2, 2, 2, 4, 2, 0, 4, 4, 4, 0, 1, 2, 0, 4, 2, 0, 2, 2, 5, 2, 0, 2, 2, 4, 4, 2, 0, 2, 4, 2, 2, 0, 4, 0, 0, 2, 3, 2, 4, 2, 0, 4, 0, 6, 2, 4, 1, 0, 4, 2, 2, 2, 2, 0, 0, 2, 0, 2, 8, 2, 2, 0, 2, 4, 0, 4, 2, 2, 3, 2, 2

Offset: 0

N. J. A. Sloane

nonn

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

Number of ways to write n as an ordered sum of 2 generalized pentagonal numbers. - Ilya Gutkovskiy, Aug 14 2017

			G.f. = 1 + 2*x + 3*x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 4*x^7 + 2*x^8 + 2*x^9 + ...
G.f. = q + 2*q^13 + 3*q^25 + 2*q^37 + q^49 + 2*q^61 + 2*q^73 + 4*q^85 + 2*q^97 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Cerkan, Table of n, a(n) for n = 0..10000
J. W. L. Glaisher, On the square of Euler's series, Proc. London Math. Soc., 21 (1889), 182-194.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002654, A008441, A008442, A035154, A035181, A035184, A112301, A113406.
Cf. A113652, A116604, A121363, A121450, A122856, A122864, A122865, A125061.
Cf. A125079, A129447, A129448, A132004, A134013, A134015, A138741, A138746.
Cf. A138950, A138952, A163746, A258210, A258228, A258256, A258279, A258292.

series(mul( ( (1 + q^n)*(1 - q^(3*n))/(1 + q^(3*n)) )^2, n = 1..100), q, 101):
seq(coeftayl(%, q = 0, n), n = 0..100); # Peter Bala, Jan 05 2025

ed[n_]:=Module[{divs=Divisors[12n+1]},Count[divs,?(Mod[#,4] == 1&)]- Count[divs,?(Mod[#,4]==3&)]]; Array[ed,110,0] (* Harvey P. Dale, Jul 01 2012 *)
a[ n_] := If[ n < 0, 0, With[ {m = 12 n + 1}, Sum[ KroneckerSymbol[ 4, d], {d, Divisors[m]}]]]; (* Michael Somos, Apr 23 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^3]^2 / (QPochhammer[ x] QPochhammer[ x^6]))^2, {x, 0, n}]; (* Michael Somos, Apr 23 2014 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] / QPochhammer[ x, x^2])^2, {x, 0, n}]; (* Michael Somos, May 25 2015 *)

{a(n) = if( n<0, 0, n = 12*n + 1; sumdiv( n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Sep 19 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A) * eta(x^6 + A)))^2, n))}; /* Michael Somos, Jun 02 2012 */

Expansion of (phi(-x^3) / chi(-x))^2 in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/12) * (eta(q^2) * eta(q^3)^2 / (eta(q) * eta(q^6)))^2 in powers of q. - Michael Somos, Sep 19 2005
Euler transform of period 6 sequence [ 2, 0, -2, 0, 2, -2, ...]. - Michael Somos, Sep 19 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258279. - Michael Somos, May 25 2015
From Michael Somos, Jun 02 2012: (Start)
a(n) = A008441(3*n) = A121363(3*n) = A122865(4*n) = A122856(8*n).
a(n) = A116604(6*n) = A125079(6*n) = A129447(6*n) = A138741(6*n).
a(n) = A(12*n+1) where A = A002654, A008442, A035154, A035181, A035184, A112301, A113406, A113652, A121450, A122864, A125061, A129448, A132004, A134013, A134015, A138746, A138950, A138952, A163746.  (End)
From Michael Somos, May 25 2015: (Start)
a(n) = A258277(4*n) = A258278(8*n) = A258291(3*n).
a(n) = - A258210(12*n + 1) = A258228(12*n + 1) = A258256(12*n + 1).
2*a(n) = A258279(12*n + 1) = - A258292(12*n + 1). (End)
G.f.: (Sum_{k = -oo..oo} x^(k*(3*k-1)/2))^2. - Ilya Gutkovskiy, Aug 14 2017
G.f.: ( Product_{n >= 1} (1 + q^n)*(1 - q^(3*n))/(1 + q^(3*n)) )^2. - Peter Bala, Jan 05 2025

A258228 Expansion of f(q) * f(-q^2) * chi(q^3) in powers of q where chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, May 23 2015

sign

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

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

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, A122856, A122865, A258210, A258279, A258292.

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

a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^2 / (QPochhammer[ -q, q^6] QPochhammer[ -q^5, q^6]), {q, 0, n}];

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

Expansion of f(q)^2 * f(-q^6) / f(q, q^5) in powers of q where f(,) is Ramanujan's general theta function.
Expansion of eta(q^2)^4 * eta(q^6)^2 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 1, -3, 2, -2, 1, -4, 1, -2, 2, -3, 1, -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 A122865.
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k))^2 * (1 + x^(3*k)) / ((1 + x^(2*k)) * (1 + x^(6*k))).
a(n) = (-1)^n * A258210(n) = A258279(2*n) = A258292(2*n).
a(3*n + 1) = A122865(n). a(3*n + 2) = -2 * A122856(n). a(9*n) = A004018(n). a(9*n + 3) = a(9*n + 6) = 0.
a(4*n + 3) = 0. a(6*n + 2) = -2 * A122865(n). a(12*n + 1) = A002175(n).

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.

cp's OEIS Frontend

A002175 Excess of number of divisors of 12n+1 of form 4k+1 over those of form 4k+3.

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A258228 Expansion of f(q) * f(-q^2) * chi(q^3) in powers of q where chi(), f() are Ramanujan theta functions.

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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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Magma

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