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.

Showing 1-4 of 4 results.

A115977 Expansion of elliptic modular function lambda in powers of the nome q.

Original entry on oeis.org

16, -128, 704, -3072, 11488, -38400, 117632, -335872, 904784, -2320128, 5702208, -13504512, 30952544, -68901888, 149403264, -316342272, 655445792, -1331327616, 2655115712, -5206288384, 10049485312, -19115905536, 35867019904, -66437873664
Offset: 1

Views

Author

Michael Somos, Feb 09 2006

Keywords

Comments

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

Examples

			G.f. = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + 11488*q^5 - 38400*q^6 + 117632*q^7 - ...

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
  • J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 121.
  • A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 23, eq. (37).

Links

Crossrefs

Cf. A005798, A014972, A128692, A132136.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ InverseEllipticNomeQ @ x, {x, 0, n}];
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ ModularLambda[ Log[q] / (Pi I)], {q, 0, n}]];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 3, 0, q])^4, {q, 0, n}];
a[ n_] := SeriesCoefficient[ 1/16 (EllipticTheta[ 2, 0, q] / EllipticTheta[ 3, 0, q^2])^8, {q, 0, n}]; (* Michael Somos, May 26 2016 *)
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); 16 * polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^8, n))};

Formula

Expansion of Jacobi elliptic parameter m = k^2 = (theta_2(q) / theta_3(q))^4 in powers of the nome q.
Expansion of 16 * q * (psi(q^2) / phi(q))^4 = 16 * q * (psi(q^2) / psi(q))^8 = 16 * q * (psi(q) / phi(q))^8 = 16 * q * (psi(-q) / phi(-q^2))^8 = 16 * q / (chi(q) * chi(-q^2))^8 = 16 * q * (f(-q^4) / f(q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of 16 * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^8 in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * (1 - v)^2 - 16 * v * (1 - u).
lambda( -1 / tau ) = 1 - lambda( tau ) (see A128692).
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128692.
G.f.: 16 * q * (Product_{k>0} (1 + q^(2*k)) / (1 + q^(2*k - 1)))^8.
a(n) = 16 * A005798(n). a(n) = -(-1)^n * A014972(n) unless n=0.
a(n) = -(-1)^n * A132136(n). - Michael Somos, Jun 03 2015
Empirical: Sum_{n>=1}(exp(-2*Pi)^n*a(n)) = 17 - 12*sqrt(2). - Simon Plouffe, Feb 20 2011
a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n)) / (32 * n^(3/4)). - Vaclav Kotesovec, Apr 06 2018
The g.f. A(q) = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + ... satisfies A(q) + A(-q) = A(q)*A(-q). - Peter Bala, Sep 26 2023

A014972 Expansion of (theta_3(q) / theta_4(q) )^4 in powers of q; also of 1 / (1 - lambda(z)).

Original entry on oeis.org

1, 16, 128, 704, 3072, 11488, 38400, 117632, 335872, 904784, 2320128, 5702208, 13504512, 30952544, 68901888, 149403264, 316342272, 655445792, 1331327616, 2655115712, 5206288384, 10049485312, 19115905536, 35867019904, 66437873664
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The relation with A092877 is equivalent to eta(q^2)^24 = eta(q)^16 * eta(q^4)^8 + 16 * eta(q)^8 * eta(q^4)^16. - Michael Somos, Apr 11 2004

Examples

			G.f. = 1 + 16*q + 128*q^2 + 704*q^3 + 3072*q^4 + 11488*q^5 + 38400*q^6 + 117632*q^7 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.

Links

Crossrefs

Cf. A005798, A007248, A029845, A092877, A128692, A132136.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q])^4, {q, 0, n}]; (* Michael Somos, Jun 25 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 / (QPochhammer[ q^4] QPochhammer[ q]^2))^8, {q, 0, n}]; (* Michael Somos, Jun 25 2014 *)
nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^8,{k,0,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Aug 28 2015 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( exp( 16 * sum( k=1, (n+1)\2, sigma(2*k - 1) / (2*k - 1) * x^(2*k - 1), x * O(x^n))), n))}; /* Michael Somos, Apr 11 2004 */
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)))^8, n))}; /* Michael Somos, Apr 11 2004 */

Formula

Expansion of 1 / (1 - lambda(t)) = 1 / lambda(-1 / t) in powers of q = exp(Pi i t).
Expansion of (phi(q) / phi(-q))^4 = (phi(-q^2) / phi(-q))^8 = (phi(q) / phi(-q^2))^8 = (f(q) / f(-q))^8 = (chi(q)/ chi(-q))^8 = (psi(q) / psi(-q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (eta(q^2)^3 / (eta(q^4) * eta(q)^2))^8 in powers of q. - Michael Somos, Apr 11 2004
Euler transform of period 4 sequence [ 16, -8, 16, 0, ...]. - Michael Somos, Apr 11 2004
G.f. A(x) satisfies A(-x) = 1 / A(x). Also 0 = f(A(x), A(x^2)) where f(u, v) = (u - 1)^2 + 16 * u*v * (1 - v). - Michael Somos, Apr 11 2004
G.f.: (Product_{k>0} (1 + x^(2*k - 1)) / (1 - x^(2*k - 1)))^8 = exp( 16 * Sum_{k>0} x^(2*k - 1) * sigma(2*k - 1) / (2*k - 1)). - Michael Somos, Apr 11 2004
a(n) = 16 * A092877(n) unless n = 0. a(n) = A132136(n) unless n = 0. Convolution inverse of A128692.
Empirical : Sum_{n >=1} exp(-2*Pi)^(n-1)*(-1)^(n+1)*a(n) = -16+12*2^(1/2). - Simon Plouffe, Feb 20 2011
a(n) ~ exp(2*Pi*sqrt(n)) / (32 * n^(3/4)). - Vaclav Kotesovec, Aug 28 2015

A005798 Expansion of (theta_2(q)/theta_3(q))^4/16 in powers of q.

Original entry on oeis.org

0, 1, -8, 44, -192, 718, -2400, 7352, -20992, 56549, -145008, 356388, -844032, 1934534, -4306368, 9337704, -19771392, 40965362, -83207976, 165944732, -325393024, 628092832, -1194744096, 2241688744, -4152367104, 7599231223, -13749863984
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

When multiplied by 16, this is the q-expansion of the automorphic function lambda (see A115977) [see Erdelyi].
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = q - 8*q^2 + 44*q^3 - 192*q^4 + 718*q^5 - 2400*q^6 + 7352*q^7 - 20992*q^8 + ...

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
  • J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 121.
  • A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 23, Eq. (37).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

If initial 0 is omitted and sequence begins with a(0) = 1, then this is the convolution of A001938 with itself. G.f.s are related by a(x)=x*A001938(x)^2. Reversion of A005797.
Cf. A007248, A029845, A115977, A128692.

Programs

  • Maple
    with (numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: aa:=etr (n-> [ -8,16,-8,0] [modp(n-1,4)+1]): a:= n->aa(n-1): seq (a(n), n=0..26);  # Alois P. Heinz, Sep 08 2008
  • Mathematica
    a[ n_] := SeriesCoefficient[ InverseEllipticNomeQ[ x] / 16, {x, 0, n}]; (* Michael Somos, Jun 13 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, 0, q^(1/2)])^8, {q, 0, n}]; (* Michael Somos, May 10 2014 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^4] / QPochhammer[ -q])^8, {q, 0, n}]; (* Michael Somos, May 10 2014 *)
a[ n_] := SeriesCoefficient[ q (Product[ 1 - q^k, {k, 4, n - 1, 4}] / Product[ 1 - (-q)^k, {k, n - 1}])^8, {q, 0, n}]; (* Michael Somos, May 10 2014 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; aa = etr[Function[{n}, {-8, 16, -8, 0}[[Mod[n-1, 4]+1]]]]; a[n_] := aa[n-1]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)
  • PARI
    {a(n) = my(A, m); if( n<1, 0, m=1; A = x + O(x^2); while( m
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^8, n))}; /* Michael Somos, Jul 16 2005 */

Formula

Expansion of elliptic lambda / 16 = m / 16 = (k / 4)^2 in powers of the nome q.
Expansion of q * (psi(q) / phi(q))^8 = q * (psi(q^2) / psi(q))^8 = q * (psi(-q) / phi(-q^2))^8 = q * (chi(-q) / chi(-q^2)^2)^8 = q / (chi(q) * chi(-q^2))^8 = q / (chi(-q) * chi(q)^2)^8 = q * (psi(q^2) / phi(q))^4 = q * (f(-q^4) / f(q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jun 13 2011
Expansion of eta(q)^8 * eta(q^4)^16 / eta(q^2)^24 in powers of q.
Euler transform of period 4 sequence [-8, 16, -8, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v + 16*u*v - 32*u^2*v + 256*(u*v)^2. - Michael Somos, Mar 19 2004
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = (1 / 16) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128692. - Michael Somos, May 10 2014
G.f.: q * Product( (1 + q^(2*n)) / (1 + q^(2*n - 1)), n=1..inf )^8.
a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n))/(512*n^(3/4)). - Vaclav Kotesovec, Jul 10 2016
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 17/16 - 3*sqrt(2)/4, verified to 27000 digits (10000 terms). - Simon Plouffe, Mar 01 2021

Extensions

Definition simplified by N. J. A. Sloane, Sep 25 2011

A120030 Expansion of theta_4(q)^2*theta_4(q^2)^4 in powers of q.

Original entry on oeis.org

1, -4, -4, 32, -4, -104, 32, 192, -4, -292, -104, 480, 32, -680, 192, 832, -4, -1160, -292, 1440, -104, -1536, 480, 2112, 32, -2604, -680, 2624, 192, -3368, 832, 3840, -4, -3840, -1160, 4992, -292, -5480, 1440, 5440, -104, -6728, -1536, 7392, 480, -7592, 2112, 8832, 32, -9412, -2604
Offset: 0

Views

Author

Michael Somos, Jun 05 2006

Keywords

Comments

Number 8 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 - 4*q - 4*q^2 + 32*q^3 - 4*q^4 - 104*q^5 + 32*q^6 + 192*q^7 - 4*q^8 + ...

References

  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.7).

Links

Crossrefs

Cf. A000141, A002173, A050470, A128692.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(4), 3), 51); A[1] - 4*A[2]; /* Michael Somos, May 24 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 (QPochhammer[ q] / QPochhammer[ q^4])^2)^2, {q, 0, n}]; (* Michael Somos, May 24 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 DivisorSum[ n, #^2 KroneckerSymbol[ -4, #] &]]; (* Michael Somos, May 24 2015 *)
  • PARI
    {a(n) = if( n<1, n==0, -4 * sumdiv( n, d, d^2 * kronecker( -4, d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^2 + A)^6 / eta(x^4 + A)^4, n))};

Formula

Expansion of eta(q)^4 * eta(q^2)^6 / eta(q^4)^4 in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^2 * (u - v)^2 - 4 * u*w * (v - w) * (u - 2*v).
Euler transform of period 4 sequence [ -4, -10, -4, -6, ...].
G.f.: 1 - 4 * Sum_{k>0} A056594(k-1) * k^2 * x^k / (1 - x^k).
Expansion of phi(-q)^2 * phi(-q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Aug 15 2007
G.f.: (Sum_{k in Z} (-1)^k * x^k^2)^2 * (Sum_{k in Z} (-1)^k * x^(2*k^2))^4.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 128 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A050470.
a(n) = -4 * A002173(n) unless n=0.
Convolution of A000141 and A128692.
Showing 1-4 of 4 results.

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