A263548 -id:A263548 - cp's OEIS frontend

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.

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

A263571 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, 0, 2, 0, 0, 4, 0, 1, 0

Offset: 0

Michael Somos, Oct 21 2015

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..5000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A113780, A115660, A128581, A128582, A192013, A260089, A261115, A263548, A263577.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 0, 0, With[ {m = 3 n + 1}, Sum[ KroneckerSymbol[ 2, d] KroneckerSymbol[ -3, m/d], {d, Divisors[ m]}]]];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] EllipticTheta[ 3, 0, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)] / (2^(1/2) x^(3/8)), {x, 0, n}];

{a(n) = my(m); if( n<0, 0, m = 3*n + 1; 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^3 + A) * eta(x^4 + A)^4 * eta(x^12 + A) / (eta(x + A) * eta(x^8 + A)^2), n))};

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^3) * eta(q^4)^4 * eta(q^12) / (eta(q) * eta(q^6) * eta(q^8)^2) in powers of q.
a(n) = b(3*n + 1) where b() is multiplicative with b(2^e) = 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).
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 24^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263577.
a(n) = A115660(3*n + 1) = A192013(3*n + 1) = A128581(6*n + 2).
a(2*n) = A261115(n). a(2*n + 1) = A263548(n). a(4*n + 1) = a(n). a(4*n + 3) = 2 * A128582(n).
a(8*n + 4) = a(8*n + 6) = 0. a(8*n) = A113780(n). a(8*n + 2) = 2 * A260089(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(6) = 1.282549... . - Amiram Eldar, Dec 28 2023

A263577 Expansion of psi(-x^2) * psi(x^3)^2 / f(-x^24) in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Oct 21 2015

sign

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

			G.f. = 1 - x^2 + 2*x^3 - 2*x^5 - x^8 - 2*x^11 + 2*x^12 - 2*x^14 + 2*x^18 + ...

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. A046113, A263548, A263571.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x] EllipticTheta[ 2, 0, x^(3/2)]^2 / (2^(5/2) x QPochhammer[ x^24]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^12] EllipticTheta[ 2, Pi/4, x] / (2^(1/2) x^(1/4) QPochhammer[ x^3, -x^3]^2), {x, 0, n}];

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

Expansion of phi(-x^12) * psi(-x^2) * chi(x^3)^2 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of eta(q^2) * eta(q^6)^4 * eta(q^8) / (eta(q^3)^2 * eta(q^4) * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ 0, -1, 2, 0, 0, -3, 0, -1, 2, -1, 0, -2, 0, -1, 2, -1, 0, -3, 0, 0, 2, -1, 0, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 6^(3/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263571.
a(3*n) = A046113(n). a(3*n + 1) = 0. a(3*n + 2) = - A263548(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).

A263649 a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = -2(-1)^e if e>0, 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).

Original entry on oeis.org

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

Offset: 1

Michael Somos, Oct 22 2015

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

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

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A115660, A128580, A128582, A261115, A263548, A263571.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 4, {-1, 1, -2}[[#]] (-1)^#2, Mod[#, 24] < 12, (#2 + 1) KroneckerSymbol[ #, 12]^#2, True, 1 - Mod[#2, 2]]& @@@ FactorInteger[n])];

{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==2, (-1)^e, p==3, -2 * (-1)^e, p%24>12, 1-e%2, (e+1) * kronecker(p, 12)^e )))};

a(2*n) = - a(n). a(3*n) = 2 * A115660(n). a(3*n + 1) = A263571(n+1). a(3*n + 2) = - A263548(n).
a(6*n + 1) = A261115(n). a(6*n + 3) = 2 * A128580(n). a(6*n + 5) = -2 * A128582(n).
Sum_{k=1..n} abs(a(k)) ~ (2/3)*sqrt(2/3)*Pi*n. - Amiram Eldar, Jan 29 2024

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).

cp's OEIS Frontend

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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A263571 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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A263577 Expansion of psi(-x^2) * psi(x^3)^2 / f(-x^24) in powers of x where psi(), f() are Ramanujan theta functions.

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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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A263649 a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = -2*(-1)^e if e>0, 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).

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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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PARI

Formula

A263649 a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = -2(-1)^e if e>0, 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).