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.

A256052 Coefficients of mock modular form H_2^(4) (divided by 2).

Original entry on oeis.org

8, 24, 56, 112, 216, 392, 672, 1128, 1840, 2912, 4536, 6936, 10416, 15456, 22632, 32728, 46872, 66472, 93400, 130200, 180104, 247312, 337392, 457408, 616392, 826104, 1101352, 1460928, 1928880, 2535280, 3318000, 4324824, 5615224, 7263424, 9362088, 12025904, 15396856, 19650792
Offset: 0

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Author

N. J. A. Sloane, Mar 24 2015

Keywords

Links

Crossrefs

Equals 8 times A256209.

Programs

  • Mathematica
    nmax = 50; a:= 8*CoefficientList[Series[q*Sum[q^(k - 1)*(Product[1 + q^j, {j, 1, 2 k - 2}])/(Product[1 - q^(2 j - 1), {j, 1, k}])^2, {k, 0, nmax}], {q, 0, 150}], q]; Table[a[[n]], {n, 1, 100}] (* G. C. Greubel, Jul 27 2018 *)

A257640 Expansion of psi(x)^2 / phi(-x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 1, 4, 6, 2, 11, 14, 4, 24, 30, 10, 47, 58, 18, 88, 108, 32, 156, 188, 57, 268, 318, 94, 444, 522, 152, 716, 834, 244, 1129, 1308, 378, 1744, 2010, 576, 2652, 3038, 870, 3968, 4524, 1288, 5857, 6650, 1884, 8540, 9660, 2730, 12312, 13878, 3906, 17572
Offset: 0

Views

Author

Michael Somos, Nov 04 2015

Keywords

Comments

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

Examples

			G.f. = 1 + 2*x + x^2 + 4*x^3 + 6*x^4 + 2*x^5 + 11*x^6 + 14*x^7 + 4*x^8 + ...
G.f. = q + 2*q^5 + q^9 + 4*q^13 + 6*q^17 + 2*q^21 + 11*q^25 + 14*q^29 + ...

References

  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 3, 2nd equation.

Links

Crossrefs

Cf. A053270, A256209.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (1/4) x^(-1/4) EllipticTheta[ 2, 0, x^(1/2)]^2 / EllipticTheta[ 4, 0, x^3], {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A) / (eta(x + A)^2 * eta(x^3 + A)^2), n))};
  • PARI
    q='q+O('q^99); Vec(eta(q^2)^4*eta(q^6)/(eta(q)^2*eta(q^3)^2)) \\ Altug Alkan, Apr 21 2018

Formula

Expansion of q^(-1/4) * eta(q^2)^4 * eta(q^6) / (eta(q)^2 * eta(q^3)^2) in powers of q.
Euler transform of period 6 sequence [ 2, -2, 4, -2, 2, -1, ...].
a(n) = A053270(n) unless n == 2 (mod 3). a(3*n) = A053270(3*n) - 2 * A256209(n).

A279715 Expansion of a q-series used by Ramanujan in his Lost Notebook.

Original entry on oeis.org

1, 2, 4, 6, 10, 16, 23, 34, 50, 70, 98, 136, 184, 250, 336, 444, 586, 768, 997, 1290, 1660, 2120, 2698, 3420, 4310, 5414, 6776, 8442, 10488, 12986, 16020, 19710, 24180, 29574, 36082, 43910, 53293, 64538, 77980, 94000, 113082, 135760, 162648, 194502, 232164
Offset: 0

Views

Author

Michael Somos, Dec 17 2016

Keywords

Examples

			G.f. = 1 + 2*x + 4*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 23*x^6 + 34*x^7 + ...

References

  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=1.

Links

Crossrefs

Cf. A256209.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[x^k* (&*[(1+x^j): j in [0..k]])/(&*[(1-x^(2*s+1)): s in [0..k]]) : k in [0..(m+2)]])/2  )); // G. C. Greubel, Nov 12 2018
  • Mathematica
    a[ n_] := SeriesCoefficient[ Sum[ x^k QPochhammer[ -x, x, k] / QPochhammer[ x, x^2, k + 1] // FunctionExpand, {k, 0, n}], {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=0, n, x^k * prod(i=1, k, 1 + x^i, 1 + A) / prod(i=1, k+1, 1 - x^(2*i-1), 1 + A), A), n))};

Formula

G.f.: Sum_{k>=0} x^k * (1 + x) * (1 + x^2) * ... * (1 + x^k) / ((1 - x) * (1 - x^3) * ... * (1 - x^(2*k+1))).

A292445 Expansion of a q-series used by Ramanujan in his Lost Notebook.

Original entry on oeis.org

1, 4, 10, 22, 44, 82, 145, 248, 410, 658, 1036, 1598, 2420, 3614, 5322, 7738, 11132, 15850, 22353, 31260, 43366, 59708, 81650, 110932, 149788, 201112, 268562, 356790, 471732, 620834, 813480, 1061496, 1379626, 1786282, 2304440, 2962566, 3795921, 4848160
Offset: 0

Views

Author

Michael Somos, Sep 16 2017

Keywords

Comments

Similar to A292420 but with a=1.

Examples

			G.f. = 1 + 4*x + 10*x^2 + 22*x^3 + 44*x^4 + 82*x^5 + 145*x^6 + ...

References

  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=1.

Links

Crossrefs

Cf. A022567, A053253, A256209, A279715, A292420.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^2 / QPochhammer[ x]^2 Sum[ x^k / Product[ 1 - x^(2 i + 1), {i, 0, k}], {k, 0, n}], {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^2 * sum(k=0, n, x^k / prod(i=0, k, 1 - x^(2*i+1), 1 + A/x^k)), n))};

Formula

a(n) = 2 * A256209(n) - A279715(n).
G.f. is the product of the g.f. of A022567 and A053253.
Showing 1-4 of 4 results.

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

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