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.

A006304 Coefficients of the '2nd-order' mock theta function A(q).

Original entry on oeis.org

0, 1, 2, 3, 5, 8, 11, 16, 23, 31, 43, 58, 76, 101, 132, 170, 219, 280, 354, 447, 562, 699, 869, 1076, 1323, 1625, 1987, 2418, 2937, 3556, 4289, 5162, 6196, 7413, 8853, 10547, 12530, 14860, 17586, 20763, 24474, 28792, 33802, 39624, 46368, 54163
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The "second-order" mock theta function A(q). - Jeremy Lovejoy, Dec 19 2008

Examples

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 11*x^6 + 16*x^7 + 23*x^8 + ...

References

  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 8.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A006305, A006306.

Programs

  • Mathematica
    Series[Sum[q^(n+1)^2 Product[1+q^(2k-1), {k, 1, n}]/Product[1-q^(2k-1), {k, 1, n+1}]^2, {n, 0, 9}], {q, 0, 100}]
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k + 1)^2 QPochhammer[ -x, x^2, k] / QPochhammer[ x, x^2, k + 1]^2, {k, 0, Sqrt[ n] - 1}], {x, 0, n}]]; (* Michael Somos, Apr 08 2015 *)
nmax = 100; CoefficientList[Series[Sum[x^(k+1)^2 * Product[1 + x^(2*j - 1), {j, 1, k}] / Product[1 - x^(2*j - 1), {j, 1, k+1}]^2, {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 11 2019 *)

Formula

G.f.: Sum_{n>=0} q^(n+1) (1+q^2)(1+q^4)...(1+q^(2n))/((1-q)(1-q^3)...(1-q^(2n+1))).
G.f.: Sum_{n>=0} q^(n+1)^2 (1+q)(1+q^3)...(1+q^(2n-1))/((1-q)(1-q^3)...(1-q^(2n+1)))^2.
a(n) ~ exp(Pi*sqrt(n/2)) / (8*sqrt(n)). - Vaclav Kotesovec, Jun 11 2019

Extensions

Corrected and extended by Dean Hickerson, Dec 13 1999

A006305 Taylor series related to one in Ramanujan's Lost Notebook.

Original entry on oeis.org

1, 2, 4, 6, 10, 16, 25, 38, 58, 84, 122, 174, 244, 338, 465, 630, 850, 1136, 1508, 1988, 2608, 3398, 4408, 5688, 7306, 9342, 11900, 15090, 19070, 24008, 30122, 37666, 46955, 58348, 72302, 89338, 110094, 135316, 165912, 202924, 247632, 301508
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			G.f. = 1 + 2*x + 4*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 25*x^6 + 38*x^7 + 58*x^8 + ...

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A006304, A006306.

Programs

  • Mathematica
    Series[Sum[q^(n^2+n)/(1-q)^2 Product[(1+q^(2k))/((1-q^(2k))(1-q^(2k+1))^2), {k, 1, n}], {n, 0, 9}], {q, 0, 100}]
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k k + k) QPochhammer[ -x^2, x^2, k] / (QPochhammer[ x, x, 2 k + 1] QPochhammer[ x, x^2, k + 1] ) , {k, 0, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Jul 09 2015 *)
nmax = 100; CoefficientList[Series[Sum[x^(k^2+k)/(1-x)^2 * Product[(1+x^(2*j))/((1-x^(2*j))*(1-x^(2*j+1))^2), {j, 1, k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 11 2019 *)

Formula

G.f.: Sum_{n>=0} q^(n^2+n) (1+q^2)(1+q^4)...(1+q^(2n))/((1-q)^2 (1-q^2) (1-q^3)^2 (1-q^4) ... (1-q^(2n)) (1-q^(2n+1))^2).
a(n) ~ c * exp(r*sqrt(n)) / n^(3/4), where r = 2.74858241446108527... and c = 0.1051685561271293027... - Vaclav Kotesovec, Jun 12 2019

Extensions

Corrected and extended by Dean Hickerson, Dec 13 1999

A153140 Coefficients of the second order mock theta function B(q).

Original entry on oeis.org

1, 2, 4, 6, 9, 14, 20, 28, 40, 54, 72, 98, 129, 168, 220, 282, 360, 460, 580, 728, 912, 1134, 1404, 1734, 2129, 2604, 3180, 3864, 4680, 5658, 6812, 8182, 9808, 11718, 13968, 16618, 19720, 23350, 27600, 32550, 38313
Offset: 0

Views

Author

Jeremy Lovejoy, Dec 19 2008

Keywords

Links

Crossrefs

Other '2nd order' mock theta functions are at A006304, A006306.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(k^2+k) * Product[(1+x^(2*j))/(1-x^(2*j+1))^2, {j, 0, k}], {k, 0, Floor[Sqrt[nmax]]}]/2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)
  • PARI
    lista(nn) = my(q = qq + O(qq^nn)); gf = sum(n = 0, nn, q^n * prod(k = 1, n, 1 + q^(2*k-1)) / prod(k = 0, n, 1 - q^(2*k+1))); Vec(gf) \\ Michel Marcus, Jun 18 2013

Formula

G.f.: Sum_{n >= 0} q^(n^2+n)(1+q^2)(1+q^4)...(1+q^(2n))/(1-q)^2(1-q^3)^2...(1-q^(2n+1))^2.
G.f.: Sum_{n >= 0} q^n(1+q)(1+q^3)...(1+q^(2n-1))/(1-q)(1-q^3)...(1-q^(2n+1)).
a(n) ~ exp(Pi*sqrt(n/2)) / (2^(5/2) * sqrt(n)). - Vaclav Kotesovec, Jun 12 2019

Extensions

More terms from Michel Marcus, Jun 18 2013
Showing 1-3 of 3 results.

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

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