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 1-3 of 3 results.

A195130 Series inversion of A100130.

Original entry on oeis.org

1, 24, 852, 35744, 1645794, 80415216, 4094489992, 214888573248, 11542515402255, 631467591949480, 35063515239394764, 1971043639046131296, 111949770626330347638, 6414671157989386260432, 370360217892318010055832
Offset: 1

Views

Author

Michael Somos, Sep 11 2011

Keywords

Examples

			G.f. = q + 24*q^2 + 852*q^3 + 35744*q^4 + 1645794*q^5 + 80415216*q^6 + ...

Links

Crossrefs

Cf. A005149, A100130.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[ q / QPochhammer[ -q, q^2]^24, {q, 0, n}], q], {q, 0, n}]];
  • PARI
    {a(n) = if( n<1, 0, polcoeff( serreverse( x * prod( k=1, n, 1 + (-x)^k, 1 + x * O(x^n) )^24), n))};

Formula

A005149(n) = -(-1)^n * a(n).

A002470 Glaisher's function W(n).

Original entry on oeis.org

0, 1, 4, -8, -48, 10, 224, 80, -448, -231, 40, -248, 1408, 1466, -2240, -80, 1280, -4766, -924, 1944, -480, 9600, 6944, -2704, -8704, -15525, 5864, -3984, -14080, 25498, 2240, 10816, 33792, -29760, -19064, 800, 11088, 1994, -54432, -11728, -4480
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J. W. L. Glaisher, On the representation of a number as a sum of 14 and 16 squares, Quart. J. Math. 38 (1907), 178-236 (see p. 190).
  • 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).

Links

Crossrefs

Cf. A004018, A100130.
Bisections are A002286, A002287.

Formula

Coefficients of the q-series rho^7*(eta(q) * eta(q^4) / eta(q^2)^2)^24, where rho is A004018 and the second factor is given by A100130.

Extensions

Edited (new offset, signs added, more terms, formula) by N. J. A. Sloane, Nov 26 2018

A124863 Expansion of 1 / chi(q)^12 in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -12, 78, -376, 1509, -5316, 16966, -50088, 138738, -364284, 913824, -2203368, 5130999, -11585208, 25444278, -54504160, 114133296, -234091152, 471062830, -931388232, 1811754522, -3471186596, 6556994502, -12222818640, 22502406793
Offset: 0

Views

Author

Michael Somos, Nov 10 2006

Keywords

Comments

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

Examples

			G.f. = 1 - 12*x + 78*x^2 - 376*x^3 + 1509*x^4 - 5316*x^5 + 16966*x^6 - 50088*x^7 + ...
G.f. = q - 12*q^3 + 78*q^5 - 376*q^7 + 1509*q^9 - 5316*q^11 + 16966*q^13 - 50088*q^15 + ...

Links

Crossrefs

Cf. A022577, A100130, A112142.

Programs

  • Mathematica
    a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ ((1 - m) m/16/q)^(-1/2), {q, 0, n}]]; (* Michael Somos, Jul 22 2011 *)
a[ n_] := SeriesCoefficient[1/Product[1 + q^k, {k, 1, n, 2}]^12, {q, 0, n}]; (* Michael Somos, Jul 22 2011, fixed by Vaclav Kotesovec, Nov 16 2017 *)
nmax = 30; CoefficientList[Series[Product[1/(1 + x^(2*k - 1))^12, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 16 2017 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) / eta(x^2 + A)^2)^12, n))};

Formula

Expansion of q^(-1/2) * (k * k') / 4 in powers of q where k is the Jacobian elliptic modulus, k' the complementary modulus and q is the nome.
Expansion of q^(-1/2) * (eta(q) * eta(q^4) / eta(q^2)^2)^12 in powers of q.
Euler transform of period 4 sequence [ -12, 12, -12, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 22 2011
G.f.: Product_{k>0} (1 + (-x)^k)^12 = Product_{k>0} 1/(1 + x^(2*k - 1))^12. [corrected by Vaclav Kotesovec, Nov 16 2017]
a(n) = (-1)^n * A022577(n). Convolution inverse of A112142. Convolution square is A100130.
G.f.: T(0), where T(k) = 1 - 1/(1 - 1/(1 - 1/(1+(-x)^(k+1))^12/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)) / (128*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
G.f.: exp(-12*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 08 2018
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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