A190611 -id:A190611 - cp's OEIS frontend

A192013 a(n) = Sum_{d|n} Kronecker(-6, d).

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

Offset: 1

Michael Somos, Jun 22 2011

			G.f. = x + x^2 + x^3 + x^4 + 2*x^5 + x^6 + 2*x^7 + x^8 + x^9 + 2*x^10 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000377, A190611.

Cf. A109017(n) = Kronecker(-6, n). - Michael Somos, Jul 22 2015

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -6, d], { d, Divisors[n]}]];

{a(n) = sumdiv( n, d, kronecker( -6, d))};

a(n)=sumdivmult(n, d, kronecker(-6, d)) \\ Charles R Greathouse IV, Dec 14 2016

Moebius transform is period 24 sequence [ 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = e+1 if p == 1, 5, 7, 11 (mod 24), a(p^e) = (1 + (-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
G.f.: Sum_{k>0} x^k * (1 + x^(4*k)) * (1 + x^(6*k)) / (1 + x^(12*k)).
Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker(-6, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - Kronecker(-6, p) * p^-s)).
a(n) = A000377(n) = A000377(2*n) = A190611(2*n). a(n) = a(2*n) = a(3*n). - Michael Somos, Jul 22 2015
0 <= a(n) <= d(n) and these bounds are sharp. - Charles R Greathouse IV, Dec 14 2016
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(6) = 1.282549... . - Amiram Eldar, Oct 17 2022

A000377 Expansion of f(-q^3) * f(-q^8) * chi(-q^12) / chi(-q) in powers of q where chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

Number 42 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 + q + q^2 + q^3 + q^4 + 2*q^5 + q^6 + 2*q^7 + q^8 + q^9 + 2*q^10 + ...

George E. Andrews, editor, P. A. MacMahon: Collected Papers Volume II, MIT Press, 1986, p. 260.
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 81, Eq. (32.5).

T. D. Noe, Table of n, a(n) for n = 0..10000
George E. Andrews, Nathan Fine 1916-1994, Notices Amer. Math. Soc., 42 (No. 6, 1995), 678-679.
Alexander Berkovich and Hamza Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, The Ramanujan Journal, Vol. 20, No. 3 (2009), pp. 375-408; arXiv preprint, arXiv:math/0611300 [math.NT], 2006-2007.
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
Michael Gilleland, Some Self-Similar Integer Sequences.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc., Vol. 348, No. 12 (1996), 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, Fine's Equation.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A129402, A190611.

A := Basis( ModularForms( Gamma1(24), 1), 102); A[1] + A[2] + A[3] + A[4] + A[5] + 2*A[6] + A[7] + 2*A[8] + A[9] + A[10] + 2*A[11] + 2*A[12]; /* Michael Somos, May 17 2015 */

a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, KroneckerSymbol[ -6, #] &]] (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[(EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^6] + EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^3]) / 2, {q, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^3] QPochhammer[ q^8] QPochhammer[ -q, q] / QPochhammer[ -q^12, q^12] , {q, 0, n}]; (* Michael Somos, May 17 2015 *)

{a(n) = if( n<1, n==0, sumdiv( n, d, kronecker( -6, d)))};

{a(n) = if( n<1, n==0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -6, p) * X)))[n])};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^8 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^24 + A)), n))};

Expansion of (phi(q) * phi(q^6) + phi(q^2) * phi(q^3)) / 2 = psi(-q^2) * psi(-q^3) * chi(-q^6) * chi(-q^12) / (chi(-q) * chi(-q^2)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos, Jan 26 2006
Expansion of eta(q^2) * eta(q^3) * eta(q^8) * eta(q^12) / (eta(q) * eta(q^24)) in powers of q.
Multiplicative with a(0) = 1, a(2^e) = a(3^e) = 1, a(p^e) = e+1 if p == 1, 5, 7, 11 (mod 24), a(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24). - Michael Somos, Jun 17 2005
Moebius transform is period 24 sequence [ 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, ...]. - Michael Somos, Jan 26 2006
Euler transform of period 24 sequence [ 1, 0, 0, 0, 1, -1, 1, -1, 0, 0, 1, -2, 1, 0, 0, -1, 1, -1, 1, 0, 0, 0, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 24^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 22 2011
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(8*k)) / (1 + x^(12*k)).
G.f.: 1 + Sum_{k>0} x^k * (1 + x^(4*k)) * (1 + x^(6*k)) / (1 + x^(12*k)). - Michael Somos, Sep 10 2005
G.f.: 1 + Sum{n = -infinity...infinity} (q^n + q^(5*n)) / (1 + q^(12*n)) (see Berkovich/Yesilyurt). - Ralf Stephan, May 14 2007
a(n) = (-1)^n * A190611(n). a(24*n + 13) = a(24*n + 17) = a(24*n + 19) = a(24*n + 23) = 0. a(2*n) = a(3*n) = a(n). a(2*n + 1) = A129402(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(6) = 1.2825... . - Amiram Eldar, Oct 23 2022

Edited by Michael Somos, Sep 10 2002

A129402 Expansion of phi(x^3) * psi(x^4) + x * phi(x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Apr 13 2007

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

			G.f = 1 + x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^7 + 2*x^10 + 3*x^12 + x^13 + 2*x^14 + ...
G.f. = q + q^3 + 2*q^5 + 2*q^7 + q^9 + 2*q^11 + 2*q^15 + 2*q^21 + 3*q^25 + q^27 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 83, Eq. (32.57).

Antti Karttunen, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000377, A128580, A128582, A113780, A190615, A257920, A259895, A260308, A261115, A261118, A261119, A261122.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, KroneckerSymbol[ -6, #] &]]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2] QPochhammer[ x^3] QPochhammer[ -x, x] QPochhammer[ x^6, -x^6], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)

{a(n) = if( n<0, 0, n = 2*n+1; sumdiv( n, d, kronecker( -6, d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^4 + A)^3 * eta(x^6 + A) * eta(x^24 + A) / (eta(x + A) * eta(x^8 + A) * eta(x^12 + A)^2), n))};

Expansion of f(x^2) * f(-x^3) / (chi(-x) * chi(x^6)) in powers of x where f(), chi() are Ramanujan theta functions.
Expansion of q^(-1/2) * eta(q^3) * eta(x^4)^3 * eta(q^6) * eta(q^24) / (eta(q) * eta(q^8) * eta(q^12)^12) in powers of q.
Euler transform of period 24 sequence [ 1, 1, 0, -2, 1, -1, 1, -1, 0, 1, 1, -2, 1, 1, 0, -1, 1, -1, 1, -2, 0, 1, 1, -2, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 5, 7, 11 (mod 24), b(p^e) = (1 + (-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 24^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A190611.
a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11) = 0. a(3*n + 1) = a(n).
a(n) = A000377(2*n + 1). a(3*n + 2) = 2 * A128582(n). a(12*n) = A113780(n).
a(n) = (-1)^n * A190615(n) = (-1)^floor( (n+1) / 2) * A128580(n). - Michael Somos, Nov 11 2015
a(2*n) = A261118(n). a(2*n + 1) = A261119(n). a(3*n) = A261115(n). - Michael Somos, Nov 11 2015
a(4*n) = A260308(n). a(4*n + 1) = A257920(n). a(4*n + 2) = 2 * A259895(n). - Michael Somos, Nov 11 2015
a(n) = - A261122(4*n + 2). - Michael Somos, Nov 11 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(6) = 1.282549... . - Amiram Eldar, Dec 28 2023

A261122 Expansion of f(-x) * f(x^4, x^8)^2 / f(-x^3, -x^9) in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Aug 09 2015

sign

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^6 + 2*x^7 + x^8 - x^9 + ...

Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000377, A129402, A190611, A190615, A261118, A261119.

a[ n_] := SeriesCoefficient[ 2^(1/2) EllipticTheta[ 4, 0, x^12]^2 EllipticTheta[ 2, Pi/4, x]^2 / (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(3/2)]), {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A) * eta(x^8 + A)^2 * eta(x^12 + A)^3 / (eta(x^3 + A) * eta(x^4 + A)^2 * eta(x^24 + A)^2), n))};

Expansion of phi(-x^12)^2 * psi(-x^2)^2 / (psi(x) * psi(-x^3)) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^6) * eta(q^8)^2 * eta(q^12)^3 / (eta(q^3) * eta(q^4)^2 * eta(q^24)^2) in powers of q.
Euler transform of period 24 sequence [ -1, -1, 0, 1, -1, -1, -1, -1, 0, -1, -1, -2, -1, -1, 0, -1, -1, -1, -1, 1, 0, -1, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 384^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261119.
a(n) = (-1)^(n + floor(n/2)) * A000377(n) = (-1)^floor(n/2) * A190611(n).
a(2*n) = A190611(n). a(2*n + 1) = - A190615(n). a(4*n) = A000377(n). a(4*n + 1) = - A261118(n). a(4*n + 2) = - A129402(n). a(4*n + 3) - A261119(n).

A258587 Expansion of f(-x, -x) * f(x^2, x^10) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Nov 06 2015

sign

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

			G.f. = 1 - 2*x + x^2 - 2*x^3 + 2*x^4 + 2*x^6 - 2*x^9 + x^10 - 4*x^11 + 2*x^14 + ...
G.f. = q^2 - 2*q^5 + q^8 - 2*q^11 + 2*q^14 + 2*q^20 - 2*q^29 + q^32 - 4*q^35 + ...

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. A128581, A128582, A190611, A263548, A263571.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ -x^2, x^12] QPochhammer[ -x^10, x^12] QPochhammer[ x^12], {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ -x^2, x^4] EllipticTheta[ 2, 0, x^3] / (2^(1/2) x^(3/4)), {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^24 + A) / (eta(x^2 + A)^2 * eta(x^8 + A) * eta(x^12 + A)), n))};

Expansion of phi(-x) * chi(x^2) * psi(-x^6) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q^(-2/3) * eta(q)^2 * eta(q^4)^2 * eta(q^6) * eta(q^24) / (eta(q^2)^2 * eta(q^8) * eta(q^12)) in powers of q.
Euler transform of period 24 sequence [ -2, 0, -2, -2, -2, -1, -2, -1, -2, 0, -2, -2, -2, 0, -2, -1, -2, -1, -2, -2, -2, 0, -2, -2, ...].
a(n) = (-1)^n * A263548(n) = A128581(3*n + 2) = A190611(3*n + 2).
a(2*n) = A263571(n). a(2*n + 1) = -2 * A128582(n).

A279947 Expansion of f(x^2, x^2) * f(-x, -x^5) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Dec 23 2016

sign

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

			G.f. = 1 - x + 2*x^2 - 2*x^3 - x^5 - 2*x^7 + 3*x^8 - 2*x^9 + 2*x^10 + ...
G.f. = q - q^4 + 2*q^7 - 2*q^10 - q^16 - 2*q^22 + 3*q^25 - 2*q^28 + ...

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. A113780, A128581, A128582, A128583, A128591, A190611, A260089, A261115, A261122, A263548, A263571.

a[ n_] := If[ n < 0, 0, With[ {m = 3 n + 1}, (-1)^n DivisorSum[ m, KroneckerSymbol[ 2, #] KroneckerSymbol[ -3, m/#] &]]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2] QPochhammer[ x^6] QPochhammer[ x, x^6] QPochhammer[ x^5, x^6], {x, 0, n}];

{a(n) = my(m); if( n<0, 0, m = 3*n + 1; (-1)^n * sumdiv( m, d, kronecker( 2, d) * kronecker( -3, m/d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^5 * eta(x^6 + A)^2 / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^8 + A)^2), n))};

{a(n) = my(A, p, e); if( n<0, 0, A = factor(3*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -(-1)^e, p==3, (-1)^e, p%24==1 || p%24==7, e+1, p%24==5 || p%24==11, (e+1)*(-1)^e, !(e%2))))};

Expansion of chi(-x) * phi(x^2) * psi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/3) * eta(q) * eta(q^4)^5 * eta(q^6)^2 / (eta(q^2)^3 * eta(q^3) * eta(q^8)^2) in powers of q.
Euler transform of period 24 sequence [ -1, 2, 0, -3, -1, 1, -1, -1, 0, 2, -1, -4, -1, 2, 0, -1, -1, 1, -1, -3, 0, 2, -1, -2, ...].
a(n) = b(3*n + 1) where b() is multiplicative with b(2^e) = -(-1)^e if e>0, b(3^e) = (-1)^e, b(p^e) = e+1 if p == 1, 7 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 5, 11 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
a(n) = (-1)^n * A263571(n) = A128581(3*n + 1) = - A190611(3*n + 1) = - A261122(6*n + 2).
a(2*n) = A261115(n).
a(2*n + 1) = - A263548(n).
a(8*n + 4) = a(8*n + 6) = 0.
a(4*n + 1) = -a(n).
a(4*n + 3) = -2 * A128582(n).
a(8*n) = A113780(n).
a(8*n + 2) = 2 * A260089(n).
a(16*n + 3) = -2 * A128583(n).
a(16*n + 7) = -2 * A128591(n).

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A192013 a(n) = Sum_{d|n} Kronecker(-6, d).

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A000377 Expansion of f(-q^3) * f(-q^8) * chi(-q^12) / chi(-q) in powers of q where chi(), f() are Ramanujan theta functions.

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Mathematica

PARI

PARI

PARI

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Extensions

A129402 Expansion of phi(x^3) * psi(x^4) + x * phi(x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.

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Mathematica

PARI

PARI

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A261122 Expansion of f(-x) * f(x^4, x^8)^2 / f(-x^3, -x^9) in powers of x where f(,) is Ramanujan's general theta function.

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Mathematica

PARI

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A258587 Expansion of f(-x, -x) * f(x^2, x^10) in powers of x where f(, ) is Ramanujan's general theta function.

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Mathematica

PARI

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A279947 Expansion of f(x^2, x^2) * f(-x, -x^5) in powers of x where f(, ) is Ramanujan's general theta function.

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Mathematica

PARI

PARI