A121363 -id:A121363 - 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

A121444 Expansion of f(x^3, x^9) * f(x, x^2) in powers of x where f(, ) is Ramanujan's general theta functions.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 0, 2, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 1, 2, 2, 1, 1, 0, 3, 0, 1, 1, 0, 2, 0, 1, 1, 2, 2, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 0, 3, 0, 0, 1, 1, 2, 1, 1, 1, 1, 3, 1, 0, 1, 0, 2, 0, 1, 1, 1, 2, 1, 0, 0, 1, 3, 2

Offset: 0

Michael Somos, Jul 30 2006

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

			G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + x^7 + x^8 + x^9 + 2*x^10 + x^11 + ...
G.f. = q^5 + q^17 + q^29 + q^41 + q^53 + 2*q^65 + q^89 + q^101 + q^113 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293 (see p. 285).
K. Saito, "Extended Affine Root Systems. V. Elliptic Eta-Products and Their Dirichlet Series", Proceedings on Moonshine and related topics (Montreal, QC, 1999), 139-161, CRM Proc. Lecture Notes, 30, Amer. Math. Soc., Providence, RI, 2001. MR1881609 (2003d:11066) See page 215.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A098098, A121363, A258210, A258291, A258832.

a[ n_] := If[ n < 0, 0, Sum[ I^d, {d, Divisors[12 n + 5]}] / (2 I)]; (* Michael Somos, Jul 25 2015 *)
a[ n_] := SeriesCoefficient[ 2 x^(3/8) QPochhammer[ x^6]^3 / (QPochhammer[ x, x^2] EllipticTheta[ 2, 0, x^(3/2)]), {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)
a[ n_] := Length @ FindInstance[ 24 n + 10 == (6 j + 3)^2 + (6 k + 1)^2 && j >= 0, {j, k}, Integers, 10^9]; (* Michael Somos, Jul 02 2015 *)
a[ n_] := If[ n < 0, 0, DivisorSum[ 12 n + 5, KroneckerSymbol[ -4, #] &] / 2]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := If[ n < 0, 0, Sum[ Boole[ Mod[d, 4] == 1] - Boole[ Mod[d, 4] == 3], {d, Divisors[12 n + 5]}] / 2]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)

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

{a(n) = if( n<0, 0, n = 12*n + 5; sumdiv(n, d, (d%4==1) - (d%4==3)) / 2)};

Expansion of f(-x^3) * f(-x^6) / chi(-x) in powers of x where chi(), f() are Ramanujan theta functions.
Expansion of q^(-5/12) * eta(q^2) * eta(q^3) * eta(q^6) / eta(q) in powers of q.
Euler transform of period 6 sequence [ 1, 0, 0, 0, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258210.
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(6*k)).
-2 * a(n) = A121363(3*n + 1).
Convolution square is A098098.
a(n) = (-1)^n * A258832(n) = A052343(3*n + 1). -a(n) = A258291(3*n + 1). 2 * a(n) = A008441(3*n + 1). - Michael Somos, Jul 02 2015
From Peter Bala, Jan 07 2021: (Start)
G.f. A(x) = Sum_{n = -oo..oo} x^n/(1 - x^(12*n + 5)). See Agarwal, p. 285, equation 6.19.
A(x^2) = Sum_{n = -oo..oo} x^(2*n)/(1 - x^(12*n + 5)). Cf. A033761. (End)

A129448 Expansion of q * psi(-q) * chi(q^3)^2 * psi(-q^9) in powers of q where psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Apr 16 2007

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

Number 50 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = q - q^2 + q^4 - 2*q^5 - q^8 + 2*q^10 + 2*q^13 + q^16 - 2*q^17 - 2*q^20 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 44.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002175, A091400, A121363, A121444, A122856, A122865.

A := Basis( ModularForms( Gamma1(36), 1), 82); A[2] - A[3] + A[5] - 2*A[6] - A[9] + 2*A[11] + 2*A[14] + A[17] - 2*A[18]; /* Michael Somos, Jul 09 2015 */

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ 12, d] KroneckerSymbol[ -3, n/d], {d, Divisors[ n]}]]; (* Michael Somos, Jul 09 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q^3, q^6]^2 EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(9/2)] / (2 q^(5/4)), {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker( 12, d) * kronecker( -3, n/d)))};

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, p==2, (-1)^e, p%12>6, !(e%2), (-1)^(e * (p%12==5)) * (e+1))))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^4 * eta(x^9 + A) * eta(x^36 + A) / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2 * eta(x^18 + A)), n))};

Expansion of eta(q) * eta(q^4) * eta(q^6)^4 * eta(q^9) * eta(q^36) / (eta(q^2) * eta(q^3)^2 * eta(q^12)^2 * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ -1, 0, 1, -1, -1, -2, -1, -1, 0, 0, -1, -1, -1, 0, 1, -1, -1, -2, -1, -1, 1, 0, -1, -1, -1, 0, 0, -1, -1, -2, -1, -1, 1, 0, -1, -2, ...].
a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = 0^e, a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12), a(p^e) = e+1 if p == 1 (mod 12), a(p^e) = (-1)^e * (e+1) if p == 5 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{k>0} Kronecker(12, k) * x^k/ (1 + x^k + x^(2*k)).
|a(n)| = A091400(n). a(3*n) = a(4*n + 3) = 0. a(2*n) = -a(n). a(3*n + 1) = A122865(n). a(3*n + 2) = - A122856(n). a(4*n + 1) = A121363(n). a(12*n + 1) = A002175(n). a(12*n + 5) = -2 * A121444(n).

A208955 Expansion of phi(x) * phi(x^9) / chi(x^3)^2 in powers of x where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Mar 03 2012

sign

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

Number 74 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 + 2*x - 2*x^3 - 2*x^4 + 3*x^6 + 2*x^7 - 2*x^9 - 2*x^10 + x^12 + ...
G.f. = q + 2*q^5 - 2*q^13 - 2*q^17 + 3*q^25 + 2*q^29 - 2*q^37 - 2*q^41 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 44.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A121363.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^9] / QPochhammer[ -x^3, x^6]^2 , {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)
a[ n_] := If[ n < 0, 0, (-1)^n With[ {m = 4 n + 1}, DivisorSum[ m, KroneckerSymbol[ 12, #] KroneckerSymbol[ 12, m/#] &]]]; (* Michael Somos, Jun 09 2015 *)

{a(n) = my(m); if( n<0, 0, m = 4*n + 1; (-1)^n * dirmul( vector(m, k, kronecker( 12, k)), vector(m, k, kronecker( 12, k)))[m])};

{a(n) = my(A, p, e); if( n<0, 0, n = 4*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%4==3, (1 + (-1)^e)/2, (e+1) * if( p%24>12, (-1)^e, 1)))) };

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^2 * eta(x^12 + A)^2 * eta(x^18 + A)^5 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^9 + A) * eta(x^36 + A))^2, n))};

Expansion of q^(-1/4) * eta(q^2)^5 * eta(q^3)^2 * eta(q^12)^2 * eta(q^18)^5 / (eta(q) * eta(q^4) * eta(q^6)^2 * eta(q^9) * eta(q^36))^2 in powers of q.
Euler transform of period 36 sequence  [ 2, -3, 0, -1, 2, -1, 2, -1, 2, -3, 2, -1, 2, -3, 0, -1, 2, -4, 2, -1, 0, -3, 2, -1, 2, -3, 2, -1, 2, -1, 2, -1, 0, -3, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 24 (t/i) f(t) where q = exp(2 Pi i t).
a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1, 5 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 13, 17 (mod 24).
a(n) = (-1)^n * A121363(n). a(3*n + 2) = 0.

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

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A121444 Expansion of f(x^3, x^9) * f(x, x^2) in powers of x where f(, ) is Ramanujan's general theta functions.

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A129448 Expansion of q * psi(-q) * chi(q^3)^2 * psi(-q^9) in powers of q where psi(), chi() are Ramanujan theta functions.

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A208955 Expansion of phi(x) * phi(x^9) / chi(x^3)^2 in powers of x where phi(), chi() are Ramanujan theta functions.

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