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 1-6 of 6 results.

A336975 Expansion of Product_{k>=1} 1/(1 - x^k * (k + x)).

Original entry on oeis.org

1, 1, 4, 9, 22, 47, 107, 221, 468, 953, 1932, 3814, 7560, 14625, 28192, 53757, 101827, 190907, 356362, 659716, 1215314, 2224968, 4053914, 7346367, 13260001, 23822114, 42629786, 75991017, 134991954, 238948942, 421656911, 741750026, 1301116634, 2275985891, 3971022904
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2020

Keywords

Links

Crossrefs

Cf. A006906, A227681, A336976, A336977, A336978, A336979, A336980.

Programs

  • Mathematica
    m = 34; CoefficientList[Series[Product[1/(1 - x^k*(k + x)), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 01 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(1/prod(k=1, N, 1-x^k*(k+x)))
  • PARI
    N=66; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, (k/d+x)^d/d))))

Formula

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (k/d + x)^d / d).
a(n) ~ c * n * phi^(n+1) / 5, where c = Product_{k>=3} 1/(1 - 1/phi^k*(k + 1/phi)) = 167.5661037860673786430316975350024960626825333609486463342... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 06 2021

A336976 Expansion of Product_{k>=1} 1/(1 - x^k * (1 + k*x)).

Original entry on oeis.org

1, 1, 3, 7, 15, 32, 65, 131, 260, 501, 965, 1825, 3419, 6326, 11652, 21230, 38405, 69015, 123334, 218980, 386809, 679757, 1189360, 2071761, 3594325, 6211826, 10698409, 18363038, 31420994, 53605525, 91198970, 154746133, 261929303, 442310873, 745264674, 1253081340, 2102754561
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2020

Keywords

Links

Crossrefs

Cf. A227681, A336975, A336977, A336978, A336979, A336980.

Programs

  • Mathematica
    m = 36; CoefficientList[Series[Product[1/(1 - x^k*(1 + k*x)), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 01 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(1/prod(k=1, N, 1-x^k*(1+k*x)))
  • PARI
    N=66; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, (1+k/d*x)^d/d))))

Formula

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (1 + k/d * x)^d / d).
a(n) ~ c * phi^(n+1) / sqrt(5), where c = Product_{k>=2} 1/(1 - 1/phi^k*(1 + k/phi)) = 167.5661037860673786430316975350024960626825333609486463342... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 06 2021

A336978 Expansion of Product_{k>=1} (1 - x^k * (1 + k*x)).

Original entry on oeis.org

1, -1, -2, -2, 0, 3, 8, 11, 9, 8, -10, -31, -57, -58, -107, -85, -4, 120, 167, 383, 616, 905, 948, 479, -82, -125, -905, -3661, -5937, -8247, -8807, -7756, -6249, -8147, -3525, 8330, 30748, 54740, 82660, 85406, 86083, 109681, 148897, 148077, 81288, -57885, -257092, -490304
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2020

Keywords

Links

Crossrefs

Convolution inverse of A336976.
Cf. A306565, A336975, A336978, A336979, A336980.

Programs

  • Mathematica
    m = 47; CoefficientList[Series[Product[1 - x^k*(1 + k*x), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 01 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-x^k*(1+k*x)))
  • PARI
    N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, x^k*sumdiv(k, d, (1+k/d*x)^d/d))))

Formula

G.f.: exp( - Sum_{k>=1} x^k * Sum_{d|k} (1 + k/d * x)^d / d).

A336979 Expansion of Product_{k>=1} (1 + x^k * (k + x)).

Original entry on oeis.org

1, 1, 3, 6, 11, 21, 37, 69, 108, 192, 312, 522, 827, 1297, 2032, 3240, 4982, 7569, 11508, 17107, 25696, 38340, 57080, 83298, 121373, 175653, 253455, 364307, 523650, 747487, 1063375, 1498471, 2106317, 2955154, 4124071, 5750547, 8000706, 11104596, 15324290, 21093106
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2020

Keywords

Links

Crossrefs

Cf. A022629, A160571, A336975, A336976, A336977, A336978, A336980.

Programs

  • Mathematica
    m = 39; CoefficientList[Series[Product[1 + x^k*(k + x), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 01 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, 1+x^k*(k+x)))
  • PARI
    N=66; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, (-1)^(d+1)*(k/d+x)^d/d))))

Formula

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (-1)^(d+1) * (k/d + x)^d / d).

A336977 Expansion of Product_{k>=1} (1 - x^k * (k + x)).

Original entry on oeis.org

1, -1, -3, -2, 1, 9, 11, 15, 6, -18, -46, -54, -115, -101, 32, 82, 182, 455, 804, 915, 434, -114, 196, -974, -3507, -6913, -7555, -5081, -4480, -7463, -4861, 7677, 25779, 56080, 76015, 51021, 53996, 104670, 114132, 93920, -33843, -233380, -491936, -658774, -597698, -601576
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2020

Keywords

Links

Crossrefs

Convolution inverse of A336975.
Cf. A022661, A306565, A336976, A336978, A336979, A336980.

Programs

  • Mathematica
    m = 45; CoefficientList[Series[Product[1 - x^k*(k + x), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 01 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-x^k*(k+x)))
  • PARI
    N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, x^k*sumdiv(k, d, (k/d+x)^d/d))))

Formula

G.f.: exp( - Sum_{k>=1} x^k * Sum_{d|k} (k/d + x)^d / d).

A336989 Expansion of Product_{k>=1} (1 + x^k / (1 - k*x)).

Original entry on oeis.org

1, 1, 2, 5, 13, 36, 107, 343, 1184, 4391, 17448, 74082, 335131, 1610301, 8191728, 43973853, 248305235, 1470474074, 9107950029, 58856529464, 395914407606, 2766669954699, 20047716439541, 150384068021507, 1166037568730402, 9332538119883810, 77004693701288392, 654279226353488820
Offset: 0

Views

Author

Seiichi Manyama, Aug 10 2020

Keywords

Crossrefs

Cf. A126348, A336980, A336990, A336991.

Programs

  • Mathematica
    m = 27; CoefficientList[Series[Product[1 + x^k/(1 - k*x), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, Aug 10 2020 *)
  • PARI
    N=40; x='x+O('x^N); Vec(prod(k=1, N, 1+x^k/(1-k*x)))
  • PARI
    N=40; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, (-1)^(d+1)/(d*(1-k/d*x)^d)))))

Formula

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (-1)^(d+1) / (d * (1 - k/d * x)^d)).
Showing 1-6 of 6 results.

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