A261119 -id:A261119 - cp's OEIS frontend

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.

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

A257921 Expansion of f(x^2, -x^4) * f(-x, -x^5)^2 / f(-x^12, -x^12) in powers of x where f(, ) is Ramanujan's general theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jul 12 2015

sign

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

			G.f. = 1 - 2*x + 2*x^2 - 2*x^3 + x^6 - 2*x^7 + 4*x^8 - 4*x^13 + 2*x^14 + ...
G.f. = q^3 - 2*q^7 + 2*q^11 - 2*q^15 + q^27 - 2*q^31 + 4*q^35 - 4*q^55 + ...

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. A115660, A128580, A128591, A134177, A257920, A259668, A259896, A261119.

a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 3}, DivisorSum[ m, KroneckerSymbol[ 12, #] KroneckerSymbol[ -2, m/#] &]]];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)])^2 / (2^(5/2) x^(3/4) EllipticTheta[ 2, Pi/4, x] EllipticTheta[ 4, 0, x^12]), {x, 0, n};

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

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

Expansion of psi(-x)^2 * psi(x^3)^2 / (psi(-x^2) * phi(-x^12)) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(-3/4) * eta(q)^2 * eta(q^4)^3 * eta(q^6)^4 * eta(q^24) / (eta(q^2)^3 * eta(q^3)^2 *eta(q^8) * eta(q^12)^2) in powers of q.
Euler transform of period 24 sequence [ -2, 1, 0, -2, -2, -1, -2, -1, 0, 1, -2, -2, -2, 1, 0, -1, -2, -1, -2, -2, 0, 1, -2, -2, ...].
a(n) = b(4*n + 3) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 11 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 5, 7 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
a(n) = (-1)^n * A261119(n) = A134177(2*n + 1) = - A128580(2*n + 1) = - A115660(4*n+3).
a(6*n + 4) = a(6*n + 5) = 0.
a(2*n) = A257920(n). a(2*n + 1) = -2 * A259896(n). a(3*n) = A259668(n). a(6*n + 2) = 2 * A128591(n).

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).

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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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A257921 Expansion of f(x^2, -x^4) * f(-x, -x^5)^2 / f(-x^12, -x^12) in powers of x where f(, ) is Ramanujan's general theta functions.

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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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