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.

A299105 Expansion of 1/(1 - x*Product_{k>=1} (1 - x^k)).

Original entry on oeis.org

1, 1, 0, -2, -3, -1, 5, 10, 7, -9, -29, -30, 10, 77, 108, 22, -184, -351, -207, 372, 1041, 969, -516, -2835, -3655, -284, 6990, 12190, 5977, -14957, -37044, -30994, 24144, 103374, 122409, -7715, -262704, -420585, -162274, 589068, 1309674, 972747, -1057935, -3742955
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 02 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A286354.
Cf. similar sequences: A067687, A299106, A299208, A302017, A318581, A318582, A331484.
Cf. A010815, A299108.

Programs

  • Mathematica
    nmax = 43; CoefficientList[Series[1/(1 - x Product[1 - x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 43; CoefficientList[Series[1/(1 - x QPochhammer[x, x]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} (1 - x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A010815(k-1)*a(n-k).

A299208 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)).

Original entry on oeis.org

1, 1, 0, -1, -2, -1, 1, 3, 3, 1, -3, -6, -5, 1, 9, 12, 5, -9, -20, -18, 1, 26, 38, 21, -21, -61, -62, -9, 72, 120, 81, -44, -177, -205, -64, 186, 366, 293, -63, -496, -657, -304, 445, 1084, 1014, 33, -1341, -2053, -1238, 959, 3132, 3378, 770, -3474, -6260, -4619, 1656, 8809, 10929, 4306, -8520
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A286352.
Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318581, A318582, A331484.
Cf. A081362, A299108, A299162, A299164, A299166, A299167, A299209, A299210, A299211, A299212.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[1/(1 - x Product[1/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A081362(k-1)*a(n-k).

A307058 Expansion of 1/(2 - Product_{k>=1} (1 + x^(2*k-1))).

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 12, 21, 38, 68, 120, 212, 377, 670, 1188, 2107, 3740, 6638, 11778, 20898, 37084, 65808, 116775, 207212, 367696, 652478, 1157815, 2054524, 3645730, 6469316, 11479734, 20370656, 36147506, 64143372, 113821732, 201975429, 358403220, 635982680, 1128544452, 2002589998
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 21 2019

Keywords

Comments

Invert transform of A000700.

Links

Crossrefs

Cf. A000700, A055887, A302017, A304969, A307059.
Row sums of A341279.

Programs

  • Magma
    m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[1 + x^(2*j-1): j in [1..m+2]])) )); // G. C. Greubel, Jan 24 2024
  • Maple
    g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*g(i), i=1..n))
    end:
seq(a(n), n=0..39);  # Alois P. Heinz, Feb 09 2021
  • Mathematica
    nmax = 39; CoefficientList[Series[1/(2 - Product[(1 + x^(2 k - 1)), {k, 1, nmax}]), {x, 0, nmax}], x]
  • SageMath
    m=80;
def f(x): return 1/(2 - product(1+x^(2*j-1) for j in range(1,m+3)))
def A307058_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307058_list(m) # G. C. Greubel, Jan 24 2024

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A000700(k)*a(n-k).
From G. C. Greubel, Jan 24 2024: (Start)
G.f.: (1+x)/(2*(1+x) - x*QPochhammer(-1/x; x^2)).
G.f.: 1/( 2 - x^(1/24)*etx(x^2)^2/(eta(x^4)*eta(x)) ), where eta(x) is the Dedekind eta function. (End)

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.
Showing 1-6 of 6 results.

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