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 1-4 of 4 results.

A307311 Expansion of Product_{k>=1} 1/(1 + x^k/(1 - x)^k).

Original entry on oeis.org

1, -1, -1, -2, -3, -4, -5, -6, -6, -1, 19, 74, 200, 461, 977, 1987, 3976, 7902, 15559, 30105, 56778, 103833, 183765, 314882, 523007, 841752, 1305431, 1916607, 2540433, 2609983, 381628, -8814988, -36463325, -109113400, -285322360, -689608522, -1579574566, -3477967848
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 02 2019

Keywords

Comments

First differences of the binomial transform of A081362.
Convolution inverse of A129519.

Crossrefs

Programs

  • Magma
    m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(&*[(1+x^k/(1-x)^k): k in [1..m+2]]) )); // G. C. Greubel, Apr 03 2019
    
  • Maple
    a:=series(mul(1/(1+x^k/(1-x)^k),k=1..100), x=0,38): seq(coeff(a, x, n), n=0..37); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1 + x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
  • PARI
    m=40; my(x='x+O('x^m)); Vec( 1/prod(k=1,m+2, (1+x^k/(1-x)^k)) ) \\ G. C. Greubel, Apr 03 2019
    
  • Sage
    m=40; (1/product(1+x^k/(1-x)^k for k in (1..m+2))).series(x, m).coefficients(x, sparse=False) # G. C. Greubel, Apr 03 2019

A307548 Expansion of Product_{k>=1} (1 - (x/(1+x))^k).

Original entry on oeis.org

1, -1, 0, 1, -2, 4, -9, 21, -48, 105, -218, 429, -803, 1442, -2521, 4380, -7734, 14091, -26468, 50405, -94980, 172824, -296704, 467589, -644459, 678109, -177123, -1752141, 7003180, -19432494, 46778567, -104623822, 224830880, -473859273, 992825436, -2084921584
Offset: 0

Views

Author

Seiichi Manyama, Apr 14 2019

Keywords

Crossrefs

Convolution inverse of A320590.

Programs

  • Mathematica
    m = 35; CoefficientList[Series[Product[1 - (x/(1+x))^k, {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-(x/(1+x))^k))

Formula

O.g.f.: Sum_{n >= 0} (-1)^n * x^(n*(n+1)/2)/Product_{k = 1..n} ((1 + x)^k - x^k). Cf. A320591. - Peter Bala, Dec 22 2020

A327671 Expansion of Product_{k>=1} (1 - (x*(1 - x))^k).

Original entry on oeis.org

1, -1, 0, 2, -1, 1, -5, 11, -17, 26, -36, 35, -22, 19, -67, 219, -480, 687, -469, -573, 2508, -4785, 6370, -6445, 5235, -4543, 8681, -26815, 75043, -173159, 334721, -563200, 876876, -1363232, 2208921, -3621971, 5631540, -7897299, 9738858, -10479294, 9989646, -9350820
Offset: 0

Views

Author

Seiichi Manyama, Sep 21 2019

Keywords

Crossrefs

Convolution inverse of A307500.

Programs

  • Mathematica
    m = 41; CoefficientList[Series[Product[(1 - (x*(1 - x))^k), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 07 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-(x*(1-x))^k))
    
  • PARI
    N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, sigma(k)*(x*(1-x))^k/k)))

Formula

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

A307574 Expansion of Product_{k>=1} (1 - (x/(1-x))^k)^k.

Original entry on oeis.org

1, -1, -3, -6, -10, -11, 3, 63, 240, 677, 1622, 3415, 6277, 9485, 8917, -9299, -83683, -309568, -902995, -2315518, -5411355, -11662530, -23117627, -41317787, -62820880, -65358588, 29550902, 449154266, 1783671567, 5453429052, 14668699694, 36273441659
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2019

Keywords

Crossrefs

Convolution inverse of A320563.

Programs

  • Mathematica
    m = 31; CoefficientList[Series[Product[(1 - (x/(1-x))^k)^k, {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-(x/(1-x))^k)^k))
Showing 1-4 of 4 results.