cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-18 of 18 results.

A208589 Expansion of phi(x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 0, 0, 1, -2, 0, 0, -1, 4, 0, 0, 0, -6, 0, 0, 1, 8, 0, 0, 0, -12, 0, 0, -1, 18, 0, 0, -1, -24, 0, 0, 2, 32, 0, 0, 1, -44, 0, 0, -2, 58, 0, 0, -1, -76, 0, 0, 2, 100, 0, 0, 1, -128, 0, 0, -3, 164, 0, 0, -1, -210, 0, 0, 4, 264, 0, 0, 2, -332, 0, 0, -5, 416
Offset: 0

Views

Author

Michael Somos, Feb 29 2012

Keywords

Comments

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

Examples

			G.f. = 1 + 2*x + x^4 - 2*x^5 - x^8 + 4*x^9 - 6*x^13 + x^16 + 8*x^17 - 12*x^21 + ...
G.f. = 1/q + 2*q + q^7 - 2*q^9 - q^15 + 4*q^17 - 6*q^25 + q^31 + 8*q^33 + ...

Links

Crossrefs

Cf. A029838, A083365, A131125. A208850, A210063.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 2 q^(1/2) EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q^2], {q, 0, n}]; (* Michael Somos, Jul 05 2014 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)^2), n))};

Formula

Expansion of q^(1/2) * eta(q^2)^5 / (eta(q)^2 * eta(q^4) * eta(q^8)^2) in powers of q.
Given g.f. A(x), then B(q) = (A(q^2) / q)^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^2 - (v - 4) * (u - 4)^2.
Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^3 - v) * (v^3 + u) - 3*u*v * (2*(u^2 + v^2) - 11). - Michael Somos, Jul 05 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A208850. - Michael Somos, Jul 05 2014
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A029838(n). a(4*n + 1) = 2 * A083365(n).
Convolution square is A131125. Convolution inverse is A210063. - Michael Somos, Jul 05 2014

A307497 Expansion of Product_{k>=1} (1+x^k)^((-1)^k*k^k).

Original entry on oeis.org

1, -1, 5, -32, 294, -3527, 51589, -894706, 17978610, -410803143, 10517824035, -298204099693, 9273022031794, -313755862498513, 11474175971184267, -450960476552715192, 18954545423649435646, -848383466771831169101, 40285210722052785437974
Offset: 0

Views

Author

Seiichi Manyama, Apr 10 2019

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = (-1)^(n+1) * n^n, g(n) = -1.

Links

Crossrefs

Cf. A083365, A261053, A266964, A284474, A307462.

Programs

  • Mathematica
    nmax=20; CoefficientList[Series[Product[(1+x^k)^((-1)^k*k^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 12 2019 *)
  • PARI
    N=20; x='x+O('x^N); Vec(prod(k=1, N, (1+x^k)^((-1)^k*k^k)))

Formula

a(n) ~ (-1)^n * n^n * (1 + exp(-1)/n + (exp(-1)/2 + 5*exp(-2))/n^2). - Vaclav Kotesovec, Apr 12 2019

A320049 Expansion of (psi(x) / phi(x))^6 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -6, 27, -98, 309, -882, 2330, -5784, 13644, -30826, 67107, -141444, 289746, -578646, 1129527, -2159774, 4052721, -7474806, 13569463, -24274716, 42838245, -74644794, 128533884, -218881098, 368859591, -615513678, 1017596115, -1667593666, 2710062756, -4369417452
Offset: 0

Views

Author

Seiichi Manyama, Oct 04 2018

Keywords

Links

Crossrefs

(psi(x) / phi(x))^b: A083365 (b=1), A079006 (b=2), A187053 (b=3), A001938 (b=4), A195861 (b=5), this sequence (b=6), A320050 (b=7).
Cf. A029843.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[((1-x^k) * (1-x^(4*k))^2 / (1-x^(2*k))^3)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2018 *)

Formula

Convolution inverse of A029843.
Expansion of q^(-3/4) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^6 in powers of q.
a(n) ~ (-1)^n * 3^(1/4) * exp(Pi*sqrt(3*n)) / (128*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Oct 06 2018

A320050 Expansion of (psi(x) / phi(x))^7 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -7, 35, -140, 483, -1498, 4277, -11425, 28889, -69734, 161735, -362271, 786877, -1662927, 3428770, -6913760, 13660346, -26492361, 50504755, -94766875, 175221109, -319564227, 575387295, -1023624280, 1800577849, -3133695747, 5399228149, -9214458260, 15584195428
Offset: 0

Views

Author

Seiichi Manyama, Oct 04 2018

Keywords

Comments

In general, for b > 0 and (psi(x) / phi(x))^b, a(n) ~ (-1)^n * b^(1/4) * exp(Pi*sqrt(b*(n/2))) / (2^(b + 7/4) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2018

Links

Crossrefs

(psi(x) / phi(x))^b: A083365 (b=1), A079006 (b=2), A187053 (b=3), A001938 (b=4), A195861 (b=5), A320049 (b=6), this sequence (b=7).
Cf. A029844.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[((1-x^k) * (1-x^(4*k))^2 / (1-x^(2*k))^3)^7, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2018 *)

Formula

Convolution inverse of A029844.
Expansion of q^(-7/8) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^7 in powers of q.
a(n) ~ (-1)^n * 7^(1/4) * exp(Pi*sqrt((7*n)/2)) / (256*2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Oct 06 2018

A093085 Expansion of phi(-x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 0, 0, 1, 2, 0, 0, -1, -4, 0, 0, 0, 6, 0, 0, 1, -8, 0, 0, 0, 12, 0, 0, -1, -18, 0, 0, -1, 24, 0, 0, 2, -32, 0, 0, 1, 44, 0, 0, -2, -58, 0, 0, -1, 76, 0, 0, 2, -100, 0, 0, 1, 128, 0, 0, -3, -164, 0, 0, -1, 210, 0, 0, 4, -264, 0, 0, 2, 332, 0, 0, -5, -416, 0, 0, -2, 516, 0, 0, 5, -640, 0, 0, 2, 790, 0, 0, -6, -968
Offset: 0

Views

Author

Michael Somos, Mar 20 2004, Oct 22 2007

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
eta(q^2) * eta(q^8)^6 = eta(q)^2 * eta(q^4)^2 * eta(q^8) * eta(q^16)^2 + 2 * eta(q^2) * eta(q^4)^2 * eta(q^16)^4 is equivalent to the a(4*n), ..., a(4*n + 3) results.

Examples

			G.f. = 1 - 2*x + x^4 + 2*x^5 - x^8 - 4*x^9 + 6*x^13 + x^16 - 8*x^17 + 12*x^21 - ...
G.f. = 1/q - 2*q + q^7 + 2*q^9 - q^15 - 4*q^17 + 6*q^25 + q^31 - 8*q^33 + ...

Links

Crossrefs

Cf. A029838, A029839, A083365, A131124, A187154.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 2 x^(1/2) EllipticTheta[ 4, 0, x] / EllipticTheta[ 2, 0, x^2], {x, 0, n}];
  • PARI
    {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^[ 0, 2, 1, 2, 2, 2, 1, 2][1 + k%8], 1 + x * O(x^n)), n))}
  • PARI
    {a(n) = my(A, A2, m); if( n<0, 0, A = x + O(x^2); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = 4*A + 16*A^2 + (1 + 8*A) * sqrt(A + 4*A^2)); polcoeff( sqrt(x / A), n))}
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) / (eta(x^2 + A) * eta(x^8 + A)^2), n))}

Formula

Expansion of q^(1/2) * eta(q)^2 * eta(q^4) / (eta(q^2) * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ -2, -1, -2, -2, -2, -1, -2, 0, ...].
Given g.f. A(x), then B(q) = A(q)^2 / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u * (u + 8) * (v + 4) - v^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187154.
G.f.: Product_{k>0} (1 - x^k)^2 / ((1 - x^(4*k - 2)) * (1 - x^(8*k))^2).
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A029838(n). a(4*n + 1) = -2 * A083365(n). Convolution square is A131124. Convolution inverse is A187154.

A225915 Expansion of (k(q) / 4)^4 in powers of q where k() is a Jacobi elliptic function.

Original entry on oeis.org

1, -16, 152, -1088, 6444, -33184, 153152, -646528, 2533070, -9311664, 32387616, -107299904, 340436664, -1039026144, 3061896704, -8739810688, 24229115109, -65390485328, 172155210320, -442928464640, 1115433685796, -2753362613984, 6670224790272, -15876957230848
Offset: 2

Views

Author

Michael Somos, May 20 2013

Keywords

Examples

			G.f. = q^2 - 16*q^3 + 152*q^4 - 1088*q^5 + 6444*q^6 - 33184*q^7 + 153152*q^8 + ...

Links

Crossrefs

Cf. A001938, A005798, A079006, A083365.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (InverseEllipticNomeQ[  q] / 16)^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, 0, q^(1/2)])^16, {q, 0, n}];
a[ n_] := SeriesCoefficient[ q ( Product[ 1 - q^k, {k, 4, n - 1, 4}]/
Product[ 1 - (-q)^k, {k, n - 1}])^16, {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<2, 0, n-=2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^16, n))};

Formula

Expansion of (eta(q) * eta(q^4)^2 / eta(q^2)^3)^16 in powers of q.
Euler transform of period 4 sequence [-16, 32, -16, 0, ...].
G.f.: q^2 * (Product_{k>0} (1 + q^(2*k)) / (1 + q^(2*k - 1)))^16.
Convolution square of A005798.
a(n) ~ (-1)^n * exp(2*Pi*sqrt(2*n)) / (65536 * 2^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 17 2017

A283023 Expansion of f(-x, -x^5)^2 / (f(x^2, x^10) * f(x^6, x^18)) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, -2, 0, 2, 0, -4, 1, 6, 0, -8, 0, 12, -1, -18, 0, 24, 0, -32, 0, 44, 0, -58, 0, 76, 1, -100, 0, 128, 0, -164, 0, 210, 0, -264, 0, 332, -1, -416, 0, 516, 0, -640, -1, 790, 0, -968, 0, 1184, 2, -1444, 0, 1752, 0, -2120, 1, 2560, 0, -3078, 0, 3692, -2, -4420, 0
Offset: 0

Views

Author

Michael Somos, Feb 26 2017

Keywords

Comments

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

Examples

			G.f. = 1 - 2*x + 2*x^3 - 4*x^5 + x^6 + 6*x^7 - 8*x^9 + 12*x^11 + ...
G.f. = q^-3 - 2*q + 2*q^9 - 4*q^17 + q^21 + 6*q^25 - 8*q^33 + 12*q^41 + ...

Links

Crossrefs

Cf. A029838, A081055, A081056, A083365, A134178, A258741, A258939, A259774.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 QPochhammer[ -x^3, x^6]^2 QPochhammer[ x^12, x^24] / QPochhammer[ -x^2, x^4], {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ -x^3, x^6]^2 QPochhammer[ x^12, x^24] / EllipticTheta[ 4, 0, x^4], {x, 0, n}];
a[ n_] := SeriesCoefficient[ 2^(1/2) x^(3/4) EllipticTheta[ 4, 0, x] EllipticTheta[ 3, 0, x^3] / (EllipticTheta[ 4, 0, x^4] EllipticTheta[ 2, Pi/4, x^3]), {x, 0, n}] // Simplify;
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^4 * eta(x^8 + A) / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^4 + A)^2 * eta(x^12 + A) * eta(x^24 + A)), n))};
  • PARI
    lista(nn) = {q='q+O('q^nn); Vec(eta(q)^2*eta(q^6)^4*eta(q^8)/(eta(q^2)*eta(q^3)^2*eta(q^4)^2*eta(q^12)*eta(q^24)))} \\ Altug Alkan, Mar 21 2018

Formula

Expansion of chi(-x)^2 * chi(x^3)^2 * chi(-x^12) / chi(x^2) in powers of x where chi() is a Ramanujan theta function.
Expansion of phi(-x) * chi(x^3)^2 * chi(-x^12) / phi(-x^4) in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of phi(-x) * phi(x^3) / (phi(-x^4) * psi(-x^6)) ih powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(3/4) * eta(q)^2 * eta(q^6)^4 * eta(q^8) / (eta(q^2) * eta(q^3)^2 * eta(q^4)^2 * eta(q^12) * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [-2, -1, 0, 1, -2, -3, -2, 0, 0, -1, -2, 0, -2, -1, 0, 0, -2, -3, -2, 1, 0, -1, -2, 0, ...].
a(n) = A134178(2*n). a(6*n + 2) = A(6*n + 4) = 0.
a(2*n + 1) = -2 * A083365(n). a(4*n + 1) = -2 * A081055(n). a(4*n + 3) = 2 * A081056(n).
a(6*n) = A029838(n). a(12*n) = A258741(n). a(12*n + 6) = A259774(n). a(24*n + 12) = - A258939(n).

A306575 Expansion of 1/(1 - x - x^2/(1 - x^2 - x^3/(1 - x^3 - x^4/(1 - x^4 - x^5/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 21, 40, 77, 148, 285, 550, 1061, 2049, 3957, 7644, 14768, 28535, 55138, 106549, 205902, 397906, 768967, 1486070, 2871932, 5550233, 10726300, 20729542, 40061784, 77423250, 149628008, 289170949, 558851751, 1080037175, 2087280839, 4033881485
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 25 2019

Keywords

Crossrefs

Cf. A029838, A083365, A088352.

Programs

  • Mathematica
    nmax = 35; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-x^k, 1 - x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

Formula

a(n) ~ c * d^n, where d = 1.9326019136649450138850556203... and c = 0.389707331111778150048054243... - Vaclav Kotesovec, Jul 01 2019
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