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.

A335635 Expansion of e.g.f. Product_{k>0} 1/(1 - sin(x)^k / k).

Original entry on oeis.org

1, 1, 3, 10, 44, 215, 1252, 7992, 56024, 438341, 3672328, 32587366, 318586880, 3325053147, 35115462592, 407034567076, 5198294627456, 63965057355305, 824995119961984, 12611299833296898, 184189806819806720, 2590874864719588031, 44912343151409875456, 728583107189913021328, 11458864344772729650176
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2020

Keywords

Comments

a(30) is negative.

Crossrefs

Programs

  • Mathematica
    max = 24; Range[0, max]! * CoefficientList[Series[Product[1/(1 - Sin[x]^k/k), {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 03 2020 *)
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, 1-sin(x)^k/k)))
    
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, sin(x)^(i*j)/(i*j^i))))))

Formula

E.g.f.: exp( Sum_{i>0} Sum_{j>0} sin(x)^(i*j)/(i*j^i) ).

A335638 Expansion of e.g.f. Product_{k>0} (1 + tan(x)^k / k).

Original entry on oeis.org

1, 1, 1, 7, 22, 190, 1170, 11646, 109520, 1289168, 16018064, 223757840, 3407971488, 55709905056, 998011344928, 18778681069024, 385316251841536, 8225863823985664, 189755182485906432, 4538893733746003968, 116147781156885837824, 3078530007519830730752, 86521073899573883088896
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2020

Keywords

Crossrefs

Programs

  • Mathematica
    max = 22; Range[0, max]! * CoefficientList[Series[Product[1 + Tan[x]^k/k, {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 03 2020 *)
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1+tan(x)^k/k)))
    
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, (-1)^(i+1)*tan(x)^(i*j)/(i*j^i))))))

Formula

E.g.f.: exp( Sum_{i>0} Sum_{j>0} (-1)^(i+1)*tan(x)^(i*j)/(i*j^i) ).
Conjecture: a(n) ~ A080130 * 2^(2*n+1) * n! / Pi^(n+1). - Vaclav Kotesovec, Oct 04 2020

A335644 Expansion of e.g.f. Product_{k>0} (1 + sin(x)^k / k!).

Original entry on oeis.org

1, 1, 1, 3, 1, -23, -2, -28, -2435, 253, 118966, 158400, -5277106, -6453094, 377003877, 150562341, -38919169331, -49489639843, 4097920244054, 15989402021656, -397866849121614, -3949517739363706, 34992745696351023, 937723673130987379, -2417716098650478930, -223227071403982903362
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2020

Keywords

Crossrefs

Programs

  • Mathematica
    max = 25; Range[0, max]! * CoefficientList[Series[Product[1 + Sin[x]^k/k!, {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 04 2020 *)
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1+sin(x)^k/k!)))
    
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, (-1)^(i+1)*sin(x)^(i*j)/(i*j!^i))))))

Formula

E.g.f.: exp( Sum_{i>0} Sum_{j>0} (-1)^(i+1)*sin(x)^(i*j)/(i*(j!)^i) ).
Showing 1-3 of 3 results.