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.

A267005 Expansion of Product_{k>=1} ((1 - k*x^k) / (1 - x^k)).

Original entry on oeis.org

1, 0, -1, -2, -4, -2, -5, 6, 7, 18, 31, 50, 17, 34, 39, -122, -134, -296, -298, -556, -670, -692, 64, 26, 1119, 1054, 3953, 6804, 6645, 9250, 8644, 9702, 2817, 11386, -12415, -36382, -47757, -71782, -122506, -99540, -191083, -176382, -253722, -166706, -21024
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 08 2016

Keywords

Comments

Convolution of A022661 and A000041.

Links

Crossrefs

Cf. A000041, A022661, A267004.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1-k*x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

A268500 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 + x^k)).

Original entry on oeis.org

1, 0, 1, 2, 2, 6, 7, 14, 11, 42, 39, 70, 95, 142, 239, 378, 418, 624, 1106, 1200, 2250, 2836, 4166, 4902, 8021, 10410, 14961, 21268, 29477, 36714, 54172, 68358, 95071, 134946, 168035, 254190, 322335, 427338, 541054, 787264, 964969, 1340730, 1748094, 2311386
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 06 2016

Keywords

Links

Crossrefs

Cf. A022629, A267004, A267005, A268498, A268499.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1+k*x^k)/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x]

A267007 Expansion of Product_{k>=1} (1 + (k-1)*x^k).

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 8, 16, 20, 42, 51, 92, 132, 204, 299, 476, 644, 978, 1488, 2024, 3048, 4318, 6248, 8596, 12555, 17378, 24740, 34310, 47940, 65842, 93221, 125238, 173848, 239348, 324724, 445882, 602140, 816424, 1101096, 1495382, 1991892, 2684252, 3598248
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 08 2016

Keywords

Links

Crossrefs

Cf. A022629, A162506, A267004, A267008.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..42);  # Alois P. Heinz, Aug 15 2019
  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1+(k-1)*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = 0; Do[Do[poly[[j+1]] += (k-1)*poly[[j-k+1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly

A269144 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 - 2*x^k)).

Original entry on oeis.org

1, 3, 10, 29, 77, 195, 475, 1115, 2546, 5706, 12528, 27106, 57893, 122299, 255995, 531816, 1097377, 2252151, 4600835, 9362334, 18990645, 38418370, 77548880, 156251955, 314363615, 631703790, 1268148900, 2543812090, 5099469848, 10217529291, 20464112218
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 20 2016

Keywords

Comments

Convolution of A022629 and A070933.

Links

Crossrefs

Cf. A022629, A070933, A267004, A269153.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1+k*x^k)/(1-2*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ c * 2^n, where c = Product_{k>=1} (2^k + k)/(2^k - 1) = 19.14883592186082265751161402244824703642181055238186925199088...

A268502 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 - x^k))^k.

Original entry on oeis.org

1, 2, 8, 26, 73, 210, 558, 1460, 3663, 9090, 21846, 51690, 120140, 274480, 618656, 1374792, 3017867, 6546610, 14053312, 29852658, 62825894, 131025056, 270948160, 555811298, 1131498850, 2286780266, 4589706604, 9151298134, 18131193484, 35706460678, 69910352496
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 06 2016

Keywords

Links

Crossrefs

Cf. A000219, A022629, A266891, A267004, A268500, A268501.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[((1+k*x^k)/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]

A269339 Expansion of Product_{k>=1} (1 - k*x^k) / (1 + x^k).

Original entry on oeis.org

1, -2, -1, 0, 2, 6, -3, 12, -13, -2, -15, 2, -65, 44, 37, -90, 134, 26, 334, -270, 66, 18, 774, -1280, -15, -2266, 2627, -352, -3575, -516, -484, 5660, -3629, 21408, -20639, -1228, 15595, 31796, -22214, 55390, -104447, 58958, -160254, 180704, 17402, -103200
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 24 2016

Keywords

Links

Crossrefs

Cf. A022661, A081362, A267004, A267005, A268500.

Programs

  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1-k*x^k)/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Showing 1-6 of 6 results.

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