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.

A002103 Coefficients of expansion of Jacobi nome q in certain powers of (1/2)*(1 - sqrt(k')) / (1 + sqrt(k')).

Original entry on oeis.org

1, 2, 15, 150, 1707, 20910, 268616, 3567400, 48555069, 673458874, 9481557398, 135119529972, 1944997539623, 28235172753886, 412850231439153, 6074299605748746, 89857589279037102, 1335623521633805028
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

The Fricke reference has equation q^(1/4) = (sqrt(k) / 2) + 2(sqrt(k) / 2)^5 + 15(sqrt(k) / 2)^9 + 150(sqrt(k) / 2)^13 + 1707(sqrt(k) / 2)^17 + ... - Michael Somos, Jul 13 2013
a(n)^(1/n) tends to 16. - Vaclav Kotesovec, Jul 02 2016
a(n-1) appears in the expansion of the Jacobi nome q as q = x*Sum_{n >= 1} a(n-1)*x^(4*n) with x = (1/2)*(1 - sqrt(k')) / (1 + sqrt(k')), with the complementary modulus k' of elliptic functions. See, e.g., the Fricke, Kneser and Tricomi references, and the g.f. with example below. - Wolfdieter Lang, Jul 09 2016
The King-Canfield (1992) reference shows how this sequence is used in real life - it is one of the ingredients in solving the general quintic equation using elliptic functions. - N. J. A. Sloane, Dec 24 2019

Examples

			G.f. = 1 + 2*x + 15*x^2 + 150*x^3 + 1707*x^4 + 20910*x^5 + 268616*x^6 + 3567400*x^7 + ...
Jacobi nome q = x + 2x^5 + 15x^9 + 150x^13 + ... where x = q - 2q^5 + 5q^9 - 10q^13 + ... coefficients from A079006.
The series reversion of q = x + 2*x^5 + 15*x^9 + 150*x^13 + 1707*x^17 + ... equals (x + x^9 + x^25 + x^49 + ...)/(1 + 2*x^4 + 2*x^16 + 2*x^36 + 2*x^64 + ...).

References

  • King, R. B., and E. R. Canfield. "Icosahedral symmetry and the quintic equation." Computers & Mathematics with Applications 24.3 (1992): 13-28. See Eq. (4.28).
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • F. Tricomi, Elliptische Funktionen (German translation by M. Krafft of: Funzioni ellittiche), Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1948, p. 176, eq. (3.88).
  • Z. X. Wang and D. R. Guo, Special Functions, World Scientific Publishing, 1989, page 512.

Links

Crossrefs

Cf. A001936, A002639, A079006.

Programs

  • Mathematica
    max = 18; A079006[n_] := SeriesCoefficient[ Product[(1+x^(k+1)) / (1+x^k), {k, 1, n, 2}]^2, {x, 0, n}]; A079006[0] = 1; sq = Series[ Sum[ A079006[n]*q^(4n+1), {n, 0, max}], {q, 0, 4max}]; coes = CoefficientList[ InverseSeries[ sq, x], x]; a[n_] := coes[[4n + 2]]; Table[a[n], {n, 0, max-1}] (* Jean-François Alcover, Nov 08 2011, after Michael Somos *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (EllipticNomeQ[ 16 x] / x)^(1/4), {x, 0, n}]]; (* Michael Somos, Jul 13 2013 *)
a[ n_] := With[{m = 4 n + 1}, If[ n < 0, 0, SeriesCoefficient[ InverseSeries[ Series[ q (QPochhammer[ q^16] / QPochhammer[-q^4])^2, {q, 0, m}], x], {x, 0, m}]]]; (* Michael Somos, Jul 13 2013 *)
a[ n_] := With[{m = 4 n + 1}, SeriesCoefficient[ InverseSeries[ Series[ 1/2 EllipticTheta[ 2, 0, x^4] / EllipticTheta[ 3, 0, x^4], {x, 0, m}]], {x, 0, m}]]; (* Michael Somos, Apr 14 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, n = 4*n + 1; A = O(x^n); polcoeff( serreverse( x * (eta(x^4 + A) * eta(x^16 + A)^2 / eta(x^8 + A)^3)^2), n))};
  • PARI
    {a(n)=local(A,N=sqrtint(n)+1); A=serreverse(sum(n=1,N,x^((2*n-1)^2))/(1+2*sum(n=1,N,x^(4*n^2)) +O(x^(4*n+4)))); polcoeff(A,4*n+1)} \\ Paul D. Hanna, Jan 07 2014

Formula

a(n) = Sum {1<=k<=n} (-1)^k Sum { (4n+k)! C_1^b_1 ... C_n^b_n / (4n+1)! b_1! ... b_n! }, where the inner sum is over all partitions k = b_1 + ... + b_n, n = Sum i*b_i, b_i >= 0 and C_0=1, C_1=-2, C_2=5, C_3=-10 ... is given by (-1)^n*A001936(n).
G.f.: Series_Reversion( (theta_3(x) - theta_3(-x)) / (4*theta_3(x^4)) ) = Sum_{n>=0} a(n)*x^(4*n+1), where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). - Paul D. Hanna, Jan 07 2014

A005797 Expansion of Jacobi nome q in terms of parameter m/16.

Original entry on oeis.org

0, 1, 8, 84, 992, 12514, 164688, 2232200, 30920128, 435506703, 6215660600, 89668182220, 1305109502496, 19138260194422, 282441672732656, 4191287776164504, 62496081197436736, 935823746406530603, 14065763582458332888, 212122153814497767004, 3208590886304243284640
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

For a faster convergent series see A002103, where k' = sqrt(1 - k^2). - Wolfdieter Lang, Jul 14 2016
The Ansatz technique of A308835, A308836, and A308837 also works to produce the coefficients of this sequence from the ODE: T-d/dx(4*(1-x)*x*dT/dx)=0. - Bradley Klee, Jul 03 2019

Examples

			G.f. = x + 8*x^2 + 84*x^3 + 992*x^4 + 12514*x^5 + 164688*x^6 + 2232200*x^7 + ...
Given g.f. A(x),  then q = exp(-Pi sqrt(6)) = A( m/16 ) where m = ((2-sqrt(3))*(sqrt(3)-sqrt(2)))^2. - _Michael Somos_, Oct 30 2019

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.
  • B. C. Berndt, Ramanujan's theory of theta-functions, Theta functions: from the classical to the modern, Amer. Math. Soc., Providence, RI, 1993, pp. 1-63. MR 94m:11054.
  • C. L. Mallows, personal communication.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Reversion of A005798. Cf. A002639. Other nomes: A308835, A308836, A308837.

Programs

  • Maple
    a:= n-> coeff(series(EllipticNome(4*sqrt(x)), x, n+1), x, n):
seq(a(n), n=0..17);  # Thomas Richard, Aug 03 2022
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticNomeQ[ 16 x], {x, 0 ,n}] (* Michael Somos, Jul 11 2011 *)
  • PARI
    {a(n) = if( n < 1, 0, polcoeff( serreverse( x * prod(k=1, n-1, (1 + x^k)^(-1)^k, 1 + x * O(x^n))^8), n))} /* Michael Somos, Jul 19 2002 */
  • PARI
    {a(n) = my(A, m); if( n < 1, 0, m=1; A = x + O(x^2); while( m < n, m*=2; A = sqrt( subst(A, x, x^2)); A /= (1 + 4*A)^2); polcoeff( serreverse(A), n))} /* Michael Somos, Mar 18 2003 */

Formula

G.f.: q = q(m) = Sum_{n>=0} a(n) * (m/16)^n.
G.f.: exp( -Pi * agm(1, sqrt(1 - 16 * x) ) / agm(1, sqrt( 16*x ) ) ).

A100773 a(1) = 1, a(2) = (2*1)/1 = 2. a(n+1) = (n+1)*a(n) divided by the largest prime divisor of a(n).

Original entry on oeis.org

1, 2, 3, 4, 10, 12, 28, 32, 144, 480, 1056, 1152, 4992, 5376, 11520, 36864, 208896, 221184, 1400832, 1474560, 6193152, 19464192, 40697856, 42467328, 353894400, 1840250880, 3822059520, 21403533312, 88671780864, 91729428480, 568722456576
Offset: 1

Views

Author

Amarnath Murthy, Nov 28 2004

Keywords

Examples

			a(6) = 12, a(7) = (7*12)/3 = 28.

Links

Crossrefs

Cf. A002639.

Programs

  • Mathematica
    nxt[{n_,a_}]:={n+1,(a(n+1))/FactorInteger[a][[-1,1]]}; NestList[nxt,{1,1},30][[All,2]] (* Harvey P. Dale, Sep 20 2019 *)

Extensions

Extended by Ray Chandler, Dec 10 2004

A119349 Denominators of expansion of Jacobi nome q in parameter m.

Original entry on oeis.org

1, 16, 32, 1024, 2048, 524288, 1048576, 33554432, 67108864, 68719476736, 137438953472, 4398046511104, 8796093022208, 2251799813685248, 4503599627370496, 144115188075855872, 288230376151711744
Offset: 0

Views

Author

Eric W. Weisstein, May 15 2006

Keywords

Comments

These are 2^b(n) with b(n)= 0, 4, 5, 10, 11, 19, 20, 25, 26, 36, 37, 42, 43, 51, 52, 57, 58, 68, 69, 74, 75, 83, 84,... - R. J. Mathar, Mar 26 2025

Examples

			m/16 + m^2/32 + (21*m^3)/1024 + (31*m^4)/2048 + (6257*m^5)/524288 + ...

Links

  • Eric Weisstein's World of Mathematics, Nome

Crossrefs

Cf. A002639.
Showing 1-4 of 4 results.

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