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.

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A299209 Expansion of 1/(1 - x*Product_{k>=1} (1 - k*x^k)).

Original entry on oeis.org

1, 1, 0, -3, -6, -5, 11, 37, 59, 13, -155, -402, -415, 263, 1981, 3748, 2289, -6643, -22642, -31322, -187, 99040, 229410, 216823, -230029, -1223267, -2097812, -955237, 4468902, 13393758, 16752461, -3891704, -62382597, -131974181, -106680562, 173622424, 741553622, 1163057561, 329176545
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A297323.
Cf. A022661, A067687, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299210, A299211, A299212.

Programs

  • Mathematica
    nmax = 38; CoefficientList[Series[1/(1 - x Product[1 - k x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} (1 - k*x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A022661(k-1)*a(n-k).

A299210 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + k*x^k)).

Original entry on oeis.org

1, 1, 0, -2, -5, -3, 5, 20, 27, 17, -53, -152, -192, 31, 576, 1110, 694, -1297, -4519, -6160, -1107, 13665, 31914, 30643, -19339, -119260, -196142, -103318, 289543, 859631, 1062684, 13710, -2690348, -5675946, -4940757, 4167527, 21343918, 33874107, 16524162, -51704908, -150454546
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A297325.
Cf. A022693, A067687, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299211, A299212.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[1/(1 - x Product[1/(1 + k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + k*x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A022693(k-1)*a(n-k).

A299212 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k).

Original entry on oeis.org

1, 1, 0, -2, -5, -4, 4, 21, 35, 23, -47, -165, -239, -78, 479, 1273, 1508, -138, -4429, -9451, -8845, 6207, 37937, 67123, 45144, -83355, -308078, -455109, -166872, 873799, 2393041, 2916869, -73472, -8133572, -17828640, -17294146, 10383571, 70275162, 127401305, 90368779, -147825714
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A279928.
Cf. A067687, A255528, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299210, A299211.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[1/(1 - x Product[1/(1 + x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k).
a(0) = 1; a(n) = Sum_{k=1..n} A255528(k-1)*a(n-k).

A318581 Expansion of 1/(1 + x*Product_{k>=1} 1/(1 - x^k)).

Original entry on oeis.org

1, -1, 0, -1, 0, -1, 1, -1, 3, -1, 5, -2, 7, -7, 9, -16, 11, -29, 20, -46, 45, -66, 94, -95, 175, -161, 294, -307, 458, -594, 715, -1096, 1193, -1891, 2132, -3106, 3916, -5063, 7083, -8484, 12347, -14770, 20867, -26310, 34898, -46771, 58967, -81665, 101680, -139951, 178094, -237620
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 29 2018

Keywords

Examples

			G.f. = 1 - x - x^3 - x^5 + x^6 - x^7 + 3*x^8 - x^9 + 5*x^10 - 2*x^11 + 7*x^12 - 7*x^13 + ...

Links

Crossrefs

Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318582, A331484.
Cf. A000041, A010815.

Programs

  • Maple
    seq(coeff(series((1+x*mul((1-x^k)^(-1),k=1..n))^(-1),x,n+1), x, n), n = 0 .. 55); # Muniru A Asiru, Aug 30 2018
  • Mathematica
    nmax = 51; CoefficientList[Series[1/(1 + x Product[1/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[PartitionsP[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 51}]

Formula

G.f.: 1/(1 + x*Sum_{k>=0} A000041(k)*x^k).
a(0) = 1; a(n) = -Sum_{k=1..n} A000041(k-1)*a(n-k).

A318582 Expansion of 1/(1 + x*Product_{k>=1} (1 + x^k)).

Original entry on oeis.org

1, -1, 0, 0, -1, 1, -1, 0, 1, -1, 1, 0, 0, 1, 0, 0, 0, 1, -1, 0, 1, -3, 2, -1, -3, 4, -4, 0, 3, -5, 4, 0, -2, 4, -1, 1, 0, 3, -2, 0, 6, -11, 9, -1, -13, 18, -17, 1, 13, -23, 17, -4, -8, 13, -8, 7, -6, 15, -10, -3, 33, -50, 42, 0, -56, 85, -72, 6, 59, -100, 75, -23, -34, 53, -44, 35
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 29 2018

Keywords

Examples

			G.f. = 1 - x - x^4 + x^5 - x^6 + x^8 - x^9 + x^10 + x^13 + x^17 - x^18 + x^20 - 3*x^21 + ...

Links

Crossrefs

Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318581, A331484.
Cf. A000009, A081362.

Programs

  • Maple
    a:=series(1/(1+x*mul(1+x^k,k=1..100)),x=0,76): seq(coeff(a,x,n),n=0..75); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 75; CoefficientList[Series[1/(1 + x Product[(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[PartitionsQ[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

Formula

G.f.: 1/(1 + x*Sum_{k>=0} A000009(k)*x^k).
a(0) = 1; a(n) = -Sum_{k=1..n} A000009(k-1)*a(n-k).

A331484 Expansion of 1/(1 + x*Product_{k>=1} (1 - x^k)).

Original entry on oeis.org

1, -1, 2, -2, 3, -3, 3, -2, -1, 5, -13, 22, -36, 51, -68, 82, -86, 75, -31, -52, 201, -421, 732, -1125, 1575, -2024, 2344, -2370, 1807, -327, -2532, 7210, -14128, 23486, -35027, 47799, -59594, 66717, -63246, 41012, 10696, -104335, 252653, -465825, 746343
Offset: 0

Views

Author

Seiichi Manyama, Jan 18 2020

Keywords

Links

Crossrefs

Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318581, A318582.
Cf. A010815.

Programs

  • Mathematica
    m = 44; CoefficientList[Series[1/(1 + x*Product[1 - x^k, {k, 1, m}]), {x, 0, m}], x] (* Amiram Eldar, May 05 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(1/(1+x*prod(k=1, N, 1-x^k)))

Formula

a(0) = 1, a(n) = -Sum_{k=1..n} A010815(k-1)*a(n-k) for n > 0.
Previous Showing 11-16 of 16 results.

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