A000377 -id:A000377 - 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

A115660 Expansion of (phi(q) * phi(q^6) - phi(q^2) * phi(q^3)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Jan 28 2006

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

Number 41 of the 74 eta-quotients listed in Table I of Martin (1996). - Michael Somos, Mar 14 2012

			G.f. = q - q^2 - q^3 + q^4 - 2*q^5 + q^6 + 2*q^7 - q^8 + q^9 + 2*q^10 - 2*q^11 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, arXiv:math/0611300 [math.NT], 2006-2007.
Y. 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. A000377, A046113, A108563, A128580, A128581, A192013, A259668, A260308, A261115, A263548, A263571.

a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^4] QPochhammer[ q^6] QPochhammer[ q^24] / (QPochhammer[ q^3] QPochhammer[ q^8]), {q, 0, n}]; (* Michael Somos, Apr 19 2015 *)
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, Apr 19 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ 2, d] KroneckerSymbol[ -3, n/d], {d, Divisors[ n]}]]; (* Michael Somos, Apr 19 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1, 1, # < 5, (-1)^#2, Mod[#, 24] < 12, (#2 + 1) KroneckerSymbol[ #, 12]^#2, True, 1 - Mod[#2, 2]]& @@@ FactorInteger[n])]; (* Michael Somos, Oct 22 2015 *)

{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<5, (-1)^e, p%24<12, (e+1) * kronecker( p, 12)^e, 1-e%2)))};

{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) * eta(x^24 + A) / (eta(x^3 + A) * eta(x^8 + A)), n))};

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

Expansion of eta(q) * eta(q^4) * eta(q^6) * eta(q^24) / (eta(q^3) * eta(q^8)) 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) is multiplicative with a(2^e) = a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1, 7 (mod 24), a(p^e) = (e+1) * (-1)^e if p == 5, 11 (mod 24), a(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 / (24 t)) = 24^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{k>0} Kronecker(k,8) * x^k / (1 + x^k + x^(2*k)) = Sum_{k>0} Kronecker(k,3) * x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
abs(a(n)) = A000377(n). a(n) = (-1)^n * A128581(n). a(2*n) = a(3*n) = -a(n). a(2*n + 1) = A128580(n). - Michael Somos, Mar 14 2012
abs(a(n)) = A192013(n) unless n=0. - Michael Somos, Oct 22 2015
a(3*n + 1) = A263571(n). a(4*n) = A259668(n). a(6*n + 1) = A261115(n). a(6*n + 4) = A263548(n). a(8*n + 1) = A260308(n). - Michael Somos, Oct 22 2015
a(n) = A000377(n) - A108563(n) = A046113(n) - A000377(n). - Michael Somos, Oct 22 2015

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

A128581 Expansion of (phi(q^2) * phi(-q^3) - phi(-q) * phi(q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Mar 11 2007

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

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

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A115660, A128580.

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, KroneckerSymbol[ #, 8] KroneckerSymbol[ n/#, 3] &]]; (* Michael Somos, Nov 15 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q^3] - EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^6])/2, {q, 0, n}]; (* Michael Somos, Nov 15 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q] QPochhammer[ q^24] QPochhammer[ q^4, q^8] QPochhammer[ q^3, -q^3], {q, 0, n}]; (* Michael Somos, Nov 15 2015 *)

{a(n) = -(-1)^n * sumdiv(n, d, kronecker(d, 8) * kronecker(n/d, 3))}

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

A128581(n)={prod(i=1,matsize(n=factor(n))[1], if(12>n[i,1]%24, if(bittest(12,n[i,1]),(-1)^(n[i,1]+n[i,2]-1), if(bittest(n[i,2],0)&&n[i,1]%6>1,-n[i,2]-1,n[i,2]+1)), !bittest(n[i,2],0)))} \\ a(p^e) as given in formula. - M. F. Hasler, May 07 2018

Expansion of eta(q^2)^3 * eta(q^3) * eta(q^12) * eta(q^24) / (eta(q) * eta(q^6)^2 * eta(q^8)) in powers of q.
Euler transform of period 24 sequence [ 1, -2, 0, -2, 1, -1, 1, -1, 0, -2, 1, -2, 1, -2, 0, -1, 1, -1, 1, -2, 0, -2, 1, -2, ...].
Multiplicative with a(2^e) = -(-1)^e if e>0, a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1, 7 (mod 24), a(p^e) = (e+1)(-1)^e if p == 5, 11 (mod 24), a(p^e) = (1 + (-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
a(n) = -(-1)^n * A115660(n). a(2*n) = A115660(n). a(2*n + 1) = A128580(n).
abs(a(n)) = A000377(n) = A192013(n). - M. F. Hasler, May 07 2018

A190611 Expansion of f(q^3) * f(-q^8) * chi(-q^12) / chi(q) in powers of q where f(), chi() 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

Offset: 0

Michael Somos, May 14 2011

sign

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

			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 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000377, A129402.

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

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

{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)^3 * eta(x^8 + A) / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^24 + A)), n))};

Expansion of eta(q) * eta(q^4) * eta(q^6)^3 * eta(q^8) / (eta(q^2)^2 * eta(q^3) * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ -1, 1, 0, 0, -1, -1, -1, -1, 0, 1, -1, -2, -1, 1, 0, -1, -1, -1, -1, 0, 0, 1, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 96^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A129402.
a(n) = (-1)^n * A000377(n). a(24*n + 13) = a(24*n + 17) = a(24*n + 19) = a(24*n + 23) = 0.
a(2*n) = A000377(n). a(2*n + 1) = - A129402(n). a(3*n) = a(n). - Michael Somos, Nov 11 2015

A263548 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, 6, 2, 2, 2, 0, 0

Offset: 0

Michael Somos, Oct 20 2015

nonn

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 + ...
G.f. = q^2 + 2*q^5 + q^8 + 2*q^11 + 2*q^14 + 2*q^20 + 2*q^29 + q^32 + ...

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. A000377, A115660, A192013.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^2, x^12] QPochhammer[ -x^10, x^12] QPochhammer[ x^12], {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^2, x^4] EllipticTheta[ 2, Pi/4, 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^2 + A)^4 * eta(x^6 + A) * eta(x^24 + A) / (eta(x + 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)^4 * eta(q^6) * eta(q^24) / (eta(q)^2 * eta(q^8) * eta(q^12)) in powers of q.
Euler transform of period 24 sequence [ 2, -2, 2, -2, 2, -3, 2, -1, 2, -2, 2, -2, 2, -2, 2, -1, 2, -3, 2, -2, 2, -2, 2, -2, ...].
a(n) = A000377(3*n + 2) = A192013(3*n + 2) = A115660(6*n + 4).

A257920 Expansion of phi(x) * psi(x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jul 12 2015

nonn

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

			G.f. = 1 + 2*x + x^3 + 4*x^4 + 2*x^7 + 3*x^9 + 2*x^10 + 2*x^12 + 2*x^13 + ...
G.f. = q^3 + 2*q^11 + q^27 + 4*q^35 + 2*q^59 + 3*q^75 + 2*q^83 + 2*q^99 + ...

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. A000377, A128591, A129402, A134177, A192013.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {x, 0, n}];

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

Expansion of q^(-3/8) * eta(q^2)^5 * eta(q^6)^2 / (eta(q)^2 * eta(q^3) * eta(q^4)^2) in powers of q.
Euler transform of period 12 sequence [ 2, -3, 3, -1, 2, -4, 2, -1, 3, -3, 2, -2, ...].
a(n) = A129402(4*n + 1) = A134177(4*n + 1) = A000377(8*n + 3) = A192013(8*n + 3).
a(3*n + 2) = 0. a(3*n + 1) = 2 * A128591(n).

A259895 Expansion of psi(x^2) * psi(x^3) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jul 07 2015

nonn

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

			G.f. = 1 + x^2 + x^3 + x^5 + x^6 + 2*x^9 + x^11 + x^12 + 2*x^15 + x^18 + ...
G.f. = q^5 + q^21 + q^29 + q^45 + q^53 + 2*q^77 + q^93 + q^101 + 2*q^125 + ...

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. A000377, A128580, A128583, A129402, A134177, A190615, A192013, A257921, A259668, A259896.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (4 q^(5/8)), {x, 0, n}];
a[ n_] := If[ n < 0, 0, 1/2 Sum[ KroneckerSymbol[ -6, d], {d, Divisors[8 n + 5]}]]; (* Michael Somos, Jul 22 2015 *)

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

{a(n) = if( n<0, 0, 1/2 * sumdiv( 8*n + 5, d, kronecker( -6, d)))};

Expansion of q^(-5/8) * eta(q^4)^2 * eta(q^6)^2 / (eta(q^2) * eta(q^3)) in powers of q.
Euler transform of period 12 sequence [ 0, 1, 1, -1, 0, 0, 0, -1, 1, 1, 0, -2, ...].
a(n) = A259896(3*n + 1). a(3*n) = A128583(n). a(3*n + 1) = a(9*n + 8) = 0.
2 * a(n) = A129402(4*n + 2) = A190615(4*n + 2) = A000377(8*n + 5) = A192013(8*n + 5). - Michael Somos, Jul 22 2015
-2 * a(n) = A259668(2*n + 1) = A128580(4*n + 2) = A134177(4*n + 2) = A257921(6*n + 3). - Michael Somos, Jul 22 2015
a(3*n + 2) = A259896(n). - Michael Somos, Jul 22 2015

A046113 Coefficients in expansion of theta_3(q) * theta_3(q^6) in powers of q.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 2, 4, 0, 2, 4, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 4, 0, 2, 6, 0, 0, 4, 0, 0, 4, 0, 4, 0, 0, 2, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 2, 8, 0, 0, 4, 0, 4, 0, 0, 4, 2, 0, 0, 0, 0, 0, 8, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 4, 4, 0, 4, 0, 0

Offset: 0

N. J. A. Sloane, May 18 2002

nonn

Number of representations of n as a sum of six times a square and a square. - Ralf Stephan, May 14 2007

a(n) < 2 if and only if n is in A002480. a(n) > 0 if and only if n is in A002481. - Michael Somos, Mar 01 2011

			G.f. = 1 + 2*x + 2*x^4 + 2*x^6 + 4*x^7 + 2*x^9 + 4*x^10 + 4*x^15 + 2*x^16 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, arXiv:math/0611300 [math.NT], 2016-2017.

Cf. A000377, A002480, A002481, A115660.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^6], {q, 0, n}]; (* Michael Somos, Apr 19 2015 *)

{a(n) = my(G); if( n<0, 0, G = [ 1, 0; 0, 6]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, Mar 01 2011 */

G.f.: Sum_{ i, j = -oo..+oo } q^(i^2 + 6*j^2).
a(n) = A000377(n) + A115660(n). - Michael Somos, Mar 01 2011
a(0) = 1, a(n) = (1+(-1)^t)*b(n) for n > 0, where t is the number of prime factors of n, counting multiplicity, which are == 2,3,5,11 (mod 24), and b() is multiplicative with b(p^e) = (e+1) for primes p == 1,5,7,11 (mod 24) and b(p^e) = (1+(-1)^e)/2 for primes p == 13,17,19,23 (mod 24). (This formula is Corollary 4.2 in the Berkovich-Yesilyurt paper). - Jeremy Lovejoy, Nov 14 2024

A108563 Number of representations of n as sum of twice a square plus thrice a square.

Original entry on oeis.org

1, 0, 2, 2, 0, 4, 0, 0, 2, 0, 0, 4, 2, 0, 4, 0, 0, 0, 2, 0, 4, 4, 0, 0, 0, 0, 0, 2, 0, 4, 4, 0, 2, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 2, 0, 6, 0, 0, 4, 0, 0, 4, 0, 0, 4, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 6, 0, 8, 0, 0, 4, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 6, 4, 0, 4

Offset: 0

Ralf Stephan, May 13 2007

nonn

Number of solutions to n = 2*a^2 + 3*b^2 in integers.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) > 0 if and only if n is in A002480. a(n) < 2 if n is in A002481. - Michael Somos, Mar 01 2011

			G.f. = 1 + 2*x^2 + 2*x^3 + 4*x^5 + 2*x^8 + 4*x^11 + 2*x^12 + 4*x^14 + 2*x^18 + ...
a(0) = 1 since 0 = 2*0^2 + 3*0^2, a(5) = 4 since 5 = 2*1^2 + 3*1^2 = 2*(-1)^2 + 3*1^2 = 2*1^2 + 3*(-1)^2 = 2*(-1^2) + 3*(-1)^2.

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, arXiv:math/0611300 [math.NT], 2006-2007.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000377, A002480, A002481, A115660.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^3], {q, 0, n}]; (* Michael Somos, Apr 19 2015 *)
a[n_] := Module[{a, b, r}, r = Reduce[n == 2a^2 + 3b^2, {a, b}, Integers]; Which[r === False, 0, r[[0]] === And, 1, r[[0]] === Or, Length[r]]];
Table[a[n], {n, 0, 105}] (* Jean-François Alcover, Jan 09 2019 *)

for(n=0,120,print1(if(n<1,n==0,qfrep([2,0;0,3],n)[n]*2),","))

{a(n) = my(G); if( n<0, 0, G = [2, 0; 0, 3]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, Mar 01 2011 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^5 * eta(x^6 + A)^5 / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^8 + A)^2 * eta(x^12 + A)^2), n))}; /* Michael Somos, Jan 20 2017 */

Q = DiagonalQuadraticForm(ZZ,[3, 2])
Q.representation_number_list(102) # Peter Luschny, Jun 20 2014

G.f.: 1 + Sum_{k>0} x^k * (1 + x^(4*k)) * (1 + x^(6*k)) / (1 + x^(12*k)) - Sum_{k>0} Kronecker( k, 3) * x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
G.f.: Sum_{i, j in Z} x^(2*i^2 + 3*j^2). - Michael Somos, Mar 01 2011
Expansion of phi(q^2) * phi(q^3) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 01 2011
A115660(n) = A000377(n) - a(n). - Michael Somos, Mar 01 2011
Euler transform of period 24 sequence [0, 2, 2, -3, 0, -1, 0, -1, 2, 2, 0, -4, 0, 2, 2, -1, 0, -1, 0, -3, 2, 2, 0, -2, ...]. - Michael Somos, Jan 20 2017
Expansion of eta(q^4)^5 * eta(q^6)^5 / (eta(q^2)^2 * eta(q^3)^2 * eta(q^8)^2 * eta(q^12)^2) in powers of q. - Michael Somos, Jan 20 2017
a(0) = 1, a(n) = (1-(-1)^t)*b(n) for n > 0, where t is the number of prime factors of n, counting multiplicity, which are == 2,3,5,11 (mod 24), and b() is multiplicative with b(p^e) = (e+1) for primes p == 1,5,7,11 (mod 24) and b(p^e) = (1+(-1)^e)/2 for primes p == 13,17,19,23 (mod 24). (This formula is Corollary 4.2 in the Berkovich-Yesilyurt paper). - Jeremy Lovejoy, Nov 14 2024

Edited by Charles R Greathouse IV, Oct 28 2009

Edited by N. J. A. Sloane, Mar 04 2011

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

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A115660 Expansion of (phi(q) * phi(q^6) - phi(q^2) * phi(q^3)) / 2 in powers of q where phi() is a Ramanujan theta function.

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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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A128581 Expansion of (phi(q^2) * phi(-q^3) - phi(-q) * phi(q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.

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A190611 Expansion of f(q^3) * f(-q^8) * chi(-q^12) / chi(q) in powers of q where f(), chi() are Ramanujan theta functions.

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A263548 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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