A118271 -id:A118271 - cp's OEIS frontend

A134077 Expansion of psi(x) * phi(-x)^3 / chi(-x^3)^3 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

1, -5, 6, 8, -23, 12, 14, -30, 18, 20, -40, 24, 31, -77, 30, 32, -60, 48, 38, -70, 42, 44, -138, 48, 57, -90, 54, 72, -100, 60, 62, -184, 84, 68, -120, 72, 74, -155, 96, 80, -239, 84, 108, -150, 90, 112, -160, 120, 98, -276, 102, 104, -240, 108, 110, -190, 114

Offset: 0

Michael Somos, Oct 06 2007

sign

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 5*x + 6*x^2 + 8*x^3 - 23*x^4 + 12*x^5 + 14*x^6 - 30*x^7 + 18*x^8 + ...
G.f. = q - 5*q^3 + 6*q^5 + 8*q^7 - 23*q^9 + 12*q^11 + 14*q^13 - 30*q^15 + ...

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. A000203, A098098, A118271, A124449, A131944.

a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 4, 0, x]^3 QPochhammer[ -x^3, x^3]^3, {x, 0, n}]; (* Michael Somos, Oct 27 2015 *)
a[ n_] := SeriesCoefficient[ (1/16) x^(-1/2) (EllipticTheta[ 2, 0, x^(1/2)]^4 - 9 EllipticTheta[ 2, 0, x^(3/2)]^4), {x, 0, n}]; (* Michael Somos, Oct 27 2015 *)

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

Expansion of psi(x)^4 - 9 * x * psi(x^3)^4 in powers of x where psi() is a Ramanujan theta function.
Expansion of x^(-1/2) * (b(x)^3 * c(x^2)^2 / (3 * c(x)))^(1/2) in powers of x where b(), c() are cubic AGM functions.
Expansion of q^(-1/2) * eta(q)^5 * eta(q^6)^3 / (eta(q^2) * eta(q^3)^3) in powers of q.
Euler transform of period 6 sequence [-5, -4, -2, -4, -5, -4, ...].
a(n) = b(2*n+1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1)/(p - 1) if p>5.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 18 (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A124449
G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(2*k))^2 * (1 - x^k + x^(2*k))^3.
G.f.: Sum_{k>0} k * f(x^k) - 9 * k * f(x^(3*k)) where f(x) = x * (1 - x) / ((1 + x) * (1 + x^2)).
G.f.: f(x) - 3 * f(x^2) - 9 * f(x^3) + 2 * f(x^4) + 27 * f(x^6) - 18 * f(x^12) where f() is the g.f. of A000203.
a(n) = A131944(2*n + 1) = A118271(2*n + 1). a(3*n + 2) = 6 * A098098(n).

A185717 Expansion of q^(-1) * c(q^2) * (c(q) - c(q^4)) / 9 in powers of q^2 where c() is a cubic AGM theta function.

Original entry on oeis.org

1, 3, 6, 8, 9, 12, 14, 18, 18, 20, 24, 24, 31, 27, 30, 32, 36, 48, 38, 42, 42, 44, 54, 48, 57, 54, 54, 72, 60, 60, 62, 72, 84, 68, 72, 72, 74, 93, 96, 80, 81, 84, 108, 90, 90, 112, 96, 120, 98, 108, 102, 104, 144, 108, 110, 114, 114, 144, 126, 144, 133, 126, 156, 128

Offset: 0

Michael Somos, Feb 10 2011

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

			1 + 3*x + 6*x^2 + 8*x^3 + 9*x^4 + 12*x^5 + 14*x^6 + 18*x^7 + 18*x^8 + ...
q + 3*q^3 + 6*q^5 + 8*q^7 + 9*q^9 + 12*q^11 + 14*q^13 + 18*q^15 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A078708, A100044, A118271, A121443, A124449.

Cf. A000122, A000700, A010054, A121373.

A185717[n_] := SeriesCoefficient[(QPochhammer[q^3, q^3]/QPochhammer[-q^3, q^3])^4*(1/(QPochhammer[q, q^2]*QPochhammer[q^3, q^6])^3), {q, 0, n}];
Table[A185717[n], {n, 0, 50}] (* G. C. Greubel, Jul 10 2017 *)

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

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

Expansion of phi(-x^3)^4 / (chi(-x) * chi(-x^3))^3 in powers of x where phi(), chi() are Ramanujan theta functions.
Euler transform of period 6 sequence [ 3, 0, -2, 0, 3, -4, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = 3^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 2 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A118271.
a(3*n + 1) = 3 * a(n). A078708(2*n + 1) = A121443(2*n + 1) = A124449(2*n + 1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/9 = 1.0966227... (A100044). - Amiram Eldar, Dec 28 2023

A118272 Expansion of q^(-2/3) * (eta(q) * eta(q^3) * eta(q^6) / eta(q^2))^2 in powers of q.

Original entry on oeis.org

1, -2, 1, -4, 8, -6, 6, -8, 14, -10, 1, -16, 20, -14, 12, -16, 31, -18, 8, -20, 32, -28, 18, -24, 38, -32, 6, -28, 44, -30, 24, -40, 57, -34, 14, -36, 72, -38, 30, -48, 62, -52, 1, -44, 68, -46, 48, -56, 74, -50, 20, -64, 80, -64, 42, -56, 108, -58, 12, -60, 112, -76, 48, -64, 98, -66, 31, -80, 104, -80, 54, -88

Offset: 0

Michael Somos, Apr 21 2006

sign

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

			G.f. = 1 - 2*x + x^2 - 4*x^3 + 8*x^4 - 6*x^5 + 6*x^6 - 8*x^7 + 14*x^8 + ...
G.f. = q^2 - 2*q^5 + q^8 - 4*q^11 + 8*q^14 - 6*q^17 + 6*q^20 - 8*q^23 + ...

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. A118271, A252651.

A := Basis( ModularForms( Gamma0(36), 2), 180); A[3] - 2*A[6] + A[9]; /* Michael Somos, Mar 22 2015 */

QP:= QPochhammer; a[n_]:= SeriesCoefficient[QP[x^3]^6/(QP[-x, x^3]* QP[-x^2, x^3]*QP[x^3])^2, {x, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 15 2018 *)

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

q='q+O('q^99); Vec((eta(q)*eta(q^3)*eta(q^6)/eta(q^2))^2) \\ Altug Alkan, Apr 16 2018

Expansion of f(-x^3)^6 / f(x, x^2)^2 = phi(-x^3)^2 * f(-x, -x^5)^2 in powers of x where phi(), f() are Ramanujan theta functions. - Michael Somos, Mar 22 2015
Euler transform of period 6 sequence [ -2, 0, -4, 0, -2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 16 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A252651. - Michael Somos, Mar 22 2015
G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(3*k))^2 * (1 - x^(2*k) + x^(4*k))^2. - Michael Somos, Mar 22 2015
-3 * a(n) = A118271(3*n + 2).

A214456 Expansion of b(q^2) * (b(q) + 2 * b(q^4)) / 3 in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, -1, -3, 5, -3, -6, 15, -8, -3, 23, -18, -12, 15, -14, -24, 30, -3, -18, 69, -20, -18, 40, -36, -24, 15, -31, -42, 77, -24, -30, 90, -32, -3, 60, -54, -48, 69, -38, -60, 70, -18, -42, 120, -44, -36, 138, -72, -48, 15, -57, -93, 90, -42, -54, 231, -72, -24, 100

Offset: 0

Michael Somos, Jul 18 2012

sign

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

			1 - q - 3*q^2 + 5*q^3 - 3*q^4 - 6*q^5 + 15*q^6 - 8*q^7 - 3*q^8 + 23*q^9 + ...

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. A118271, A131943, A134077, A214361.

a[ n_] := SeriesCoefficient[ (9 EllipticTheta[ 3, 0, q^3]^4 - EllipticTheta[ 3, 0, q]^4) / 8, {q, 0, n}]
a[ n_] := SeriesCoefficient[ QPochhammer[q^2]^4 QPochhammer[-q^3, q^6]^3 /QPochhammer[-q, q^2], {q, 0, n}]
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2] QPochhammer[ -q^3, q^6])^3 EllipticTheta[ 2, 0, (-q)^(1/2)]^4 / (16 (-q)^(1/2)), {q, 0, n}]

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

{a(n) = if( n<1, n==0, -sigma(n) + if( n%3==0, 9 * sigma(n/3)) + if( n%4==0, 4 * sigma(n/4)) + if( n%12==0, -36 * sigma(n/12)))}

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); - prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 3, if( p==3, 4 - 3^(e+1), (p^(e+1) - 1) / (p - 1))))))}

Expansion of (9 * phi(q^3)^4 - phi(q)^4) / 8 = phi(q) * (psi(-q) * chi(q^3))^3 = psi(-q)^4 * (chi(q) * chi(q^3))^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^2)^2 * eta(q^4) * eta(q^6)^6 / ( eta(q^3) * eta(q^12))^3 in powers of q.
Euler transform of period 12 sequence [ -1, -3, 2, -4, -1, -6, -1, -4, 2, -3, -1, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 36 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A214361.
a(n) = -b(n) where b() is multiplicative with b(2^e) = 3 if e>0, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f.: Product_{k>0} (1 - x^(2*k))^4 * (1 + x^(6*k-3))^3 / (1 + x^(2*k-1)).
a(n) = (-1)^n * A118271(n). a(2*n) = a(4*n) = A131943(n). a(2*n + 1) = -A134077(n).

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A134077 Expansion of psi(x) * phi(-x)^3 / chi(-x^3)^3 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

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A185717 Expansion of q^(-1) * c(q^2) * (c(q) - c(q^4)) / 9 in powers of q^2 where c() is a cubic AGM theta function.

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A118272 Expansion of q^(-2/3) * (eta(q) * eta(q^3) * eta(q^6) / eta(q^2))^2 in powers of q.

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A214456 Expansion of b(q^2) * (b(q) + 2 * b(q^4)) / 3 in powers of q where b() is a cubic AGM theta function.

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