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 1-9 of 9 results.

A002611 Glaisher's function V(n).

Original entry on oeis.org

0, 1, 4, -4, -32, -16, 56, 80, 192, 98, -740, -704, 96, -224, 2440, 3520, -2624, -351, -780, -10632, 2688, 2960, -9496, 18176, 14208, -3934, 12552, -9856, -24608, -9760, -2720, -25344, -35520, 31106, 34160, 62844, 84576, 3120, -21880, -82272, 27520, -96768, -237316, 130240, -92832, 37984, 305296, -183296, 37632, 208803
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

It would be nice to have a q-series that generates this sequence. Glaisher gives many formulas but they are difficult to follow.

References

  • J. W. L. Glaisher, On the representation of a number as sum of 18 squares, Quart. J. Math. 38 (1907), 289-351 (see p. 320). [The whole 1907 volume of The Quarterly Journal of Pure and Applied Mathematics, volume 38, is freely available from Google Books]
  • 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. A002288, A004018.

Formula

a(n) = Sum_{k=1..floor(n/2)} A004018(n - 2*k) * A002288(k). - Sean A. Irvine, Mar 04 2019

Extensions

Edited and signs added by N. J. A. Sloane, Nov 26 2018
More terms from Sean A. Irvine, Mar 04 2019

A002614 Glaisher's function theta(n) (18 squares version).

Original entry on oeis.org

0, -7, 128, -975, 4608, -16340, 48384, -124303, 281600, -583746, 1146240, -2125108, 3691008, -6151880, 10055424, -15914895, 24136704, -35748899, 52583040, -75877938, 105994240, -145580124, 200279808, -272040500, 359036928, -468767690, 615599360
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

It would be nice to have a q-series that generates this sequence. Glaisher gives many formulas but they are difficult to follow.

References

  • J. W. L. Glaisher, On the representation of a number as sum of 18 squares, Quart. J. Math. 38 (1907), 289-351 (see p. 349). [The whole 1907 volume of The Quarterly Journal of Pure and Applied Mathematics, volume 38, is freely available from Google Books]
  • 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. A002288, A321546.

Formula

a(n) = (A321546(n) - A002288(n)) / 17. - Sean A. Irvine, Mar 04 2019

Extensions

Edited and signs added by N. J. A. Sloane, Nov 26 2018
More terms from Sean A. Irvine, Mar 04 2019

A034433 Expansion of q^(-3) * (eta(q) * eta(q^8))^8 in powers of q.

Original entry on oeis.org

1, -8, 20, 0, -70, 64, 56, 0, -133, -96, 148, 0, 670, -512, -968, 0, 1077, 1680, -2064, 0, -2098, 768, 4400, 0, -1766, -8128, 7044, 0, 744, 4096, -4760, 0, -9780, 16344, -6652, 0, 7894, -13440, -10320, 0, 41923, -8736, -16780, 0, -5892, -6144, 14560, 0, -27886, -11056, 55940
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			q^3 - 8*q^4 + 20*q^5 - 70*q^7 + 64*q^8 + 56*q^9 - 133*q^11 - 96*q^12 + ...

Links

Crossrefs

-8 * A002288(n) = a(4*n-3).

Programs

  • Mathematica
    QP = QPochhammer; s = (QP[q]*QP[q^8])^8 + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A) * eta(x^8 + A) )^8, n))} /* Michael Somos, Nov 11 2007 */

Formula

Euler transform of period 8 sequence [ -8, -8, -8, -8, -8, -8, -8, -16, ...]. - Michael Somos, Nov 11 2007
a(4*n+3) = 0.

A216711 Expansion of q * (phi(q) * psi(-q))^8 in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 8, 12, -64, -210, 96, 1016, 512, -2043, -1680, 1092, -768, 1382, 8128, -2520, -4096, 14706, -16344, -39940, 13440, 12192, 8736, 68712, 6144, -34025, 11056, -50760, -65024, -102570, -20160, 227552, 32768, 13104, 117648, -213360, 130752, 160526, -319520
Offset: 1

Views

Author

Michael Somos, Apr 10 2013

Keywords

Comments

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

Examples

			G.f. = q + 8*q^2 + 12*q^3 - 64*q^4 - 210*q^5 + 96*q^6 + 1016*q^7 + 512*q^8 + ...

Links

Crossrefs

Cf. A002288, A134461.

Programs

  • Magma
    A := Basis( CuspForms( Gamma0(4), 8), 39); A[1] + 8*A[2]; /* Michael Somos, Jun 10 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, q^(1/2)] / 2)^8, {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ( eta(x^2 + A)^4 / eta(x + A) / eta(x^4 + A) )^8, n))};

Formula

Expansion of (eta(q^2) / (eta(q) * eta(q^4)))^8 in powers of q.
a(n) is multiplicative with a(2) = 8, a(2^e) = -(-8)^e if e>1, a(p^e) = a(p) * a(p^(e-1)) - p^7 * a(p^(e-2)) if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 256 (t/i)^8 f(t) where q = exp(2 Pi i t).
a(n) = -(-1)^n * A002288(n). Convolution square of A134461.

A002272 Theta series of 32-dimensional Quebbemann lattice Q_32.

Original entry on oeis.org

1, 0, 0, 261120, 18947520, 535818240, 8320327680, 83347937280, 622558664640, 3614759362560, 17694184734720, 73337844372480, 272615629589760, 898646461378560, 2752654757806080, 7687895624386560
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Formula

G.f.: b(x)^8 - 192*b(x)^4*d(x) + 576*d(x)^2 where b(x) is the g.f. of A004011 and d(x) is the g.f. of A002288. - Sean A. Irvine, Jul 26 2020

A002610 Glaisher's function H'(4n+1) (18 squares version).

Original entry on oeis.org

0, 1, -6, -3, 82, -84, -444, 769, 1110, -2643, -860, 2901, -1176, 6277, 1170, -21315, -2308, 14244, 29442, 15540, -58194, -13338, -31886, 4080, 176682, -70715, -51240, 81489, -135728, 13137, -205350, 58826, 355974, 16380, 530932, -457944, -938748, 140329, 99462, 317157
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

It would be nice to have a q-series that generates this sequence. Glaisher gives many formulas but they are difficult to follow.

References

  • J. W. L. Glaisher, On the representation of a number as sum of 18 squares, Quart. J. Math. 38 (1907), 289-351 (see p. 312). [The whole 1907 volume of The Quarterly Journal of Pure and Applied Mathematics, volume 38, is freely available from Google Books]
  • 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. A002288.

Formula

See eqn. top of page 312 in Glaisher, where Theta(n) is A002288(n). - Sean A. Irvine, Mar 03 2019

Extensions

Edited and signs added by N. J. A. Sloane, Nov 26 2018
More terms from Sean A. Irvine, Mar 03 2019

A002615 Glaisher's function T_1(n).

Original entry on oeis.org

19, -145, 100, 2191, -8598, 14516, -29080, 114575, -320417, 615666, -1125492, 2139700, -3664750, 5997448, -10103304, 15992719, -23857290, 36059435, -53341900, 75622578, -105762592, 145414140, -198974280, 271923764, -359683403, 468557626
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

It would be nice to have a q-series that generates this sequence. Glaisher gives many formulas but they are difficult to follow.
Glaisher table on p. 349 apparently has typos, a(5) = -8592, a(10) = 615566. - Sean A. Irvine, Mar 04 2019

References

  • J. W. L. Glaisher, On the representation of a number as sum of 18 squares, Quart. J. Math. 38 (1907), 289-351 (see p. 349). [The whole 1907 volume of The Quarterly Journal of Pure and Applied Mathematics, volume 38, is freely available from Google Books]
  • 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. A002288, A002614.

Formula

a(n) = 19 * A002288(n) - A002614(n). - Sean A. Irvine, Mar 04 2019

Extensions

Edited and signs added by N. J. A. Sloane, Nov 26 2018
a(5) and a(10) corrected and more terms from Sean A. Irvine, Mar 04 2019

A173763 Expansion of (eta(q^2)^7 / eta(q^4)^2)^4 + 16 * (eta(q)^2 * eta(q^2) * eta(q^4)^2)^4 in powers of q.

Original entry on oeis.org

1, 16, -156, 256, 870, -2496, -952, 4096, 4653, 13920, -56148, -39936, 178094, -15232, -135720, 65536, -247662, 74448, 315380, 222720, 148512, -898368, 204504, -638976, -1196225, 2849504, 2344680, -243712, -3840450, -2171520, -1309408, 1048576, 8759088, -3962592, -828240, 1191168, 4307078
Offset: 1

Views

Author

Michael Somos, Feb 23 2010

Keywords

Comments

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

Examples

			G.f. = q + 16*q^2 - 156*q^3 + 256*q^4 + 870*q^5 - 2496*q^6 - 952*q^7 + 4096*q^8 + ...

Links

Crossrefs

Cf. A002288.

Programs

  • Magma
    Basis( CuspForms( Gamma1(2), 10), 50) [1]; /* Michael Somos, May 27 2014 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^2]^7 / QPochhammer[ q^4]^2)^4 + 16 q^2 (QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^2)^4, {q, 0, n}]; (* Michael Somos, May 28 2013 *)
  • PARI
    {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 / eta(x^4 + A)^2)^4 + 16 * x * (eta(x + A)^2 * eta(x^2 + A) * eta(x^4 + A)^2)^4, n))};
  • PARI
    q='q+O('q^99); Vec((eta(q^2)^7/eta(q^4)^2)^4+16*q*(eta(q)^2*eta(q^2)*eta(q^4)^2)^4) \\ Altug Alkan, Apr 18 2018
  • Sage
    CuspForms( Gamma1(2), 10, prec=50).0; # Michael Somos, May 28 2013

Formula

Expansion of q * (psi(q)^3 * phi(-q)^2)^4 * ((phi(q) / psi(q))^4 + 16 * q * (psi(q) / phi(q))^4) in powers of q where phi(), psi() are Ramanujan theta functions.
a(n) is multiplicative with a(2^e) = 16^e, a(p^e) = a(p) * a(p^(e-1)) - p^9 * a(p^(e-2)) if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 32 (t / i)^10 f(t) where q = exp(2 Pi i t).

A319456 a(n) = [x^n] Product_{k>=1} ((1 - x^k)*(1 - x^(2*k)))^n.

Original entry on oeis.org

1, -1, -3, 14, -11, -81, 282, -57, -2043, 5405, 2417, -46476, 94522, 110512, -943407, 1505289, 2807589, -16888311, 23645199, 46006542, -265972791, 472882620, 187884672, -3981273597, 14234579226, -19187383356, -78662039004, 502118911904, -847583768679, -2627514175002
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 19 2018

Keywords

Crossrefs

Cf. A002171, A002288, A008705, A030204, A030211, A066535, A296043, A296044, A319457.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 - x^k) (1 - x^(2 k)))^n , {k, 1, n}], {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[(QPochhammer[x] QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[Exp[n Sum[(DivisorSigma[1, 2 k] - 4 DivisorSigma[1, k]) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 29}]

Formula

a(n) = [x^n] Product_{k>=1} (1 - x^(2*k))^(2*n)/(1 + x^k)^n.
a(n) = [x^n] exp(n*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k).
Showing 1-9 of 9 results.

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

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