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.

A278399 G.f.: Re((i; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

1, -1, -1, -2, -2, -3, -2, -3, -2, -2, 0, 0, 3, 4, 8, 9, 14, 16, 22, 24, 30, 32, 39, 40, 46, 46, 51, 48, 52, 46, 46, 36, 32, 16, 8, -15, -30, -60, -82, -122, -151, -200, -238, -296, -342, -412, -464, -542, -602, -686, -750, -841, -904, -996, -1058, -1146, -1198
Offset: 0

Views

Author

Vladimir Reshetnikov, Nov 20 2016

Keywords

Comments

The q-Pochhammer symbol (a; q)inf = Product{k>=0} (1 - a*q^k).

Links

Crossrefs

Cf. A278400, A278401, A278402, A278420, A292042, A292043.

Programs

  • Maple
    G := add((-1)^n*x^(n*(2*n-1))/mul(1 - x^k, k = 1..2*n), n = 0..7):
S := series(G, x, 101):
seq(coeff(S, x, j), j = 0..100); # Peter Bala, Feb 04 2021
  • Mathematica
    Re[(QPochhammer[I, x] + O[x]^60)[[3]]]

Formula

(i; x)_inf is the g.f. for a(n) + i*A278400(n).
G.f.: Sum_{n >= 0} (-1)^n^x^(n*(2*n-1))/Product_{k = 1..2*n} 1 - x^k. - Peter Bala, Feb 04 2021

A278400 G.f.: Im((i; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

-1, -1, -1, 0, 0, 1, 2, 3, 4, 6, 6, 8, 9, 10, 10, 11, 10, 10, 8, 6, 2, 0, -7, -12, -20, -28, -39, -48, -62, -74, -90, -104, -122, -136, -156, -171, -190, -204, -222, -232, -247, -252, -260, -258, -258, -244, -232, -204, -176, -130, -84, -15, 54, 148, 244, 368
Offset: 0

Views

Author

Vladimir Reshetnikov, Nov 20 2016

Keywords

Comments

The q-Pochhammer symbol (a; q)inf = Product{k>=0} (1 - a*q^k).

Links

Crossrefs

Cf. A278399, A278401, A278402, A278420, A292042, A292043.

Programs

  • Maple
    with(gfun): series(add((-1)^(n+1)*x^(n*(2*n+1))/mul(1 - x^k, k = 1..2*n+1), n = 0..6), x, 100): seriestolist(%); # Peter Bala, Feb 06 2021
  • Mathematica
    Im[(QPochhammer[I, x] + O[x]^60)[[3]]]

Formula

(i; x)_inf is the g.f. for A278399(n) + i*a(n).
G.f.: Sum_{n >= 0} (-1)^(n+1)*x^(n*(2*n+1))/Product_{k = 1..2*n+1} 1 - x^k. - Peter Bala, Feb 06 2021

A278402 G.f.: Im(2/(i; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

1, 1, 0, -1, -1, -1, -3, -3, -2, -2, -2, -2, -1, 1, 1, 2, 5, 7, 7, 8, 11, 12, 12, 13, 15, 16, 14, 12, 12, 11, 6, 2, 1, -3, -10, -17, -21, -27, -37, -45, -50, -57, -68, -77, -81, -86, -96, -102, -101, -103, -108, -109, -103, -97, -95, -88, -71, -54, -42, -24, 5
Offset: 0

Views

Author

Vladimir Reshetnikov, Nov 20 2016

Keywords

Comments

The q-Pochhammer symbol (a; q)inf = Product{k>=0} (1 - a*q^k).

Links

Crossrefs

Cf. A278399, A278400, A278401.

Programs

  • Maple
    with(gfun): series( add( (-1)^n*x^(2*n)*(1 + x - x^(2*n+1))/mul(1 - x^k, k = 1..2*n+1), n = 0..50), x, 101): seriestolist(%); # Peter Bala, Feb 09 2021
  • Mathematica
    Im[(2/QPochhammer[I, x] + O[x]^70)[[3]]]

Formula

2/(i; x)_inf is the g.f. for A278401(n) + i*a(n).
G.f.: Sum_{n >= 0} (-1)^n*x^(2*n)*(1 + x - x^(2*n+1))/Product_{k = 1..2*n+1} (1 - x^k). - Peter Bala, Feb 09 2021

A292464 a(n) = A292136(n)^2 + A292137(n)^2.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 5, 5, 4, 4, 4, 10, 13, 13, 13, 20, 37, 37, 29, 40, 65, 72, 74, 89, 125, 178, 196, 200, 234, 325, 410, 394, 421, 617, 772, 829, 905, 1125, 1445, 1625, 1700, 2045, 2650, 3077, 3293, 3698, 4658, 5540, 5941, 6605, 7880, 9553, 10817, 11785, 13625
Offset: 0

Views

Author

Seiichi Manyama, Sep 17 2017

Keywords

Links

Crossrefs

Cf. A278401, A278402, A278420, A292136, A292137, A292138.

Programs

  • Mathematica
    Abs[CoefficientList[Series[1/QPochhammer[I*x, x], {x, 0, 100}], x]]^2 (* Vaclav Kotesovec, Sep 17 2017 *)

Formula

a(n) = A292136(n)^2 + A292138(n)^2.
a(n) = (A278401(n)^2 + A278402(n)^2)/2.
Showing 1-4 of 4 results.

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