A132217 -id:A132217 - cp's OEIS frontend

A262160 Expansion of psi(x^6) / psi(x) in powers of x where psi() is a Ramanujan theta function.

1, -1, 1, -2, 3, -4, 6, -8, 11, -15, 19, -25, 33, -42, 53, -68, 86, -107, 134, -166, 205, -253, 309, -377, 460, -557, 672, -811, 974, -1166, 1394, -1661, 1975, -2344, 2773, -3275, 3863, -4543, 5333, -6253, 7316, -8544, 9964, -11600, 13484, -15653, 18140

Offset: 0

Michael Somos, Sep 13 2015

sign

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

			G.f. = 1 - x + x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 6*x^6 - 8*x^7 + 11*x^8 + ...
G.f. = q^5 - q^13 + q^21 - 2*q^29 + 3*q^37 - 4*q^45 + 6*q^53 - 8*q^61 + ...

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. A132217.

a[ n_] := SeriesCoefficient[ x^(-5/8) EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}];

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

Expansion of q^(-5/8) * eta(q) * eta(q^12)^2 / (eta(q^2)^2 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ -1, 1, -1, 1, -1, 2, -1, 1, -1, 1, -1, 0, ...].
a(n) = (-1)^n * A132217(n).
Product_{k>0} (1 - x^(12*k)) * (1 - x^(2*k) + x^(4*k)) / (1 - (-x)^k). - Michael Somos, Oct 04 2015

A262987 Expansion of f(-x, -x^5) * f(x^3, x^5) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 1, 3, 6, 11, 19, 33, 53, 86, 134, 205, 309, 460, 672, 974, 1394, 1975, 2773, 3863, 5333, 7316, 9964, 13484, 18140, 24269, 32288, 42751, 56331, 73888, 96503, 125529, 162635, 209939, 270027, 346123, 442213, 563205, 715110, 905361, 1142998, 1439098, 1807175

Offset: 0

Michael Somos, Oct 06 2015

nonn

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

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 11*x^4 + 19*x^5 + 33*x^6 + 53*x^7 + ...
G.f. = q^5 + q^21 + 3*q^37 + 6*q^53 + 11*q^69 + 19*q^85 + 33*q^101 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A035294, A097451, A132217, A262160.

a[ n_] := SeriesCoefficient[ x^(-5/8) EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, 2 n}];
f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; a:= CoefficientList[Series[f[-x, -x^5]*f[x^3, x^5]/f[-x, -x^2]^2, {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 31 2018 *)

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

Expansion of (psi(x^6) / psi(x) + psi(x^6) / psi(-x)) / 2 in powers of x^2 where psi() is a Ramanujan theta function.
Euler transform of period 48 sequence [1, 2, 3, 2, 2, 0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 3, 1, 1, 0, 1, 1, 3, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 0, 2, 2, 3, 2, 1, 0, ...].
a(n) = A132217(2*n) = A262160(2*n).
Convolution product of A035294 and A097451.
a(n) ~ exp(sqrt(n)*Pi)/(8*sqrt(6)*n^(3/4)). - Vaclav Kotesovec, Oct 06 2015

A260162 Expansion of phi(-x) / psi(-x^3) in powers of x where psi(), phi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 0, 1, 0, 0, 1, 0, 0, 0, -2, 0, 1, -2, 0, 2, 0, 0, 1, 0, 0, 1, -2, 0, 2, -4, 0, 3, -2, 0, 2, 0, 0, 1, -4, 0, 4, -6, 0, 5, -2, 0, 3, 0, 0, 3, -6, 0, 6, -10, 0, 8, -4, 0, 5, -2, 0, 4, -10, 0, 9, -14, 0, 12, -6, 0, 8, -2, 0, 7, -14, 0, 14, -22, 0, 18, -10

Offset: 0

Michael Somos, Nov 09 2015

sign

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

			G.f. = 1 - 2*x + x^3 + x^6 - 2*x^10 + x^12 - 2*x^13 + 2*x^15 + x^18 + x^21 + ...
G.f. = 1/q^3 - 2*q^5 + q^21 + q^45 - 2*q^77 + q^93 - 2*q^101 + 2*q^117 + ...

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. A132217, A132218, A253243.

a[ n_] := SeriesCoefficient[ 2^(1/2) x^(3/8) EllipticTheta[ 4, 0, x] / EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}];
a[n_]:= SeriesCoefficient[EllipticTheta[3,0,-q]*QPochhammer[-q^3, q^6]/ QPochhammer[q^6,q^6], {q, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Mar 19 2018 *)

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

q='q+O('q^99); Vec(eta(q)^2*eta(q^6)/(eta(q^2)*eta(q^3)*eta(q^12))) \\ Altug Alkan, Mar 20 2018

Expansion of q^(3/8) * eta(q)^2 * eta(q^6) / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -2, -1, -1, -1, -2, -1, -2, -1, -1, -1, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (768 t)) = 12^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132217.
G.f.: 1 / (Product_{k>0} (1 + x^k) * (1 + x^k + x^(2*k)) * (1 + x^(6*k))).
2 * a(n) = A253243(2*n + 3). a(3*n + 2) = 0.
Convolution inverse of A132218.

A262146 Expansion of f(-x, -x^5) * f(x, x^7) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 2, 4, 8, 15, 25, 42, 68, 107, 166, 253, 377, 557, 811, 1166, 1661, 2344, 3275, 4543, 6253, 8544, 11600, 15653, 20994, 28011, 37178, 49100, 64550, 84489, 110115, 142951, 184867, 238196, 305844, 391391, 499244, 634865, 804925, 1017610, 1282957, 1613195

Offset: 0

Michael Somos, Oct 06 2015

nonn

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

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 15*x^4 + 25*x^5 + 42*x^6 + 68*x^7 + ...
G.f. = q^13 + 2*q^29 + 4*q^45 + 8*q^61 + 15*q^77 + 25*q^93 + 42*q^109 + ...

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. A078408, A097451, A132217, A262160.

a[ n_] := SeriesCoefficient[ - x^(-5/8) EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, 2 n + 1}];

{a(n) = my(A); if( n<0, 0, n = 2*n + 1; A = x * O(x^n); polcoeff( - eta(x + A) * eta(x^12 + A)^2 / (eta(x^2 + A)^2 * eta(x^6 + A)), n))};

Expansion of - (psi(x^6) / psi(x) - psi(x^6) / psi(-x)) / (2 * x) in powers of x^2 where psi() is a Ramanujan theta function.
Euler transform of period 48 sequence [ 2, 1, 2, 2, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 3, 1, 2, 0, 1, 2, 2, 2, 2, 0, 2, 2, 2, 2, 1, 0, 2, 1, 3, 1, 1, 1, 1, 2, 3, 1, 2, 1, 1, 2, 2, 1, 2, 0, ...].
a(n) = A132217(2*n + 1) = - A262160(2*n + 1).
Convolution product of A097451 and A078408.
a(n) ~ exp(Pi*sqrt(n)) / (2^(7/2) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Mar 31 2018

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A262160 Expansion of psi(x^6) / psi(x) in powers of x where psi() is a Ramanujan theta function.

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A262987 Expansion of f(-x, -x^5) * f(x^3, x^5) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.

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A260162 Expansion of phi(-x) / psi(-x^3) in powers of x where psi(), phi() are Ramanujan theta functions.

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A262146 Expansion of f(-x, -x^5) * f(x, x^7) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.

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