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.

A129519 First differences of the binomial transform of the distinct partition numbers (A000009).

Original entry on oeis.org

1, 1, 2, 5, 12, 28, 65, 151, 350, 807, 1850, 4221, 9597, 21760, 49215, 111032, 249856, 560835, 1255854, 2805969, 6256784, 13925698, 30941050, 68634679, 152009239, 336152787, 742276931, 1636747349, 3604206106, 7926412320, 17410413153
Offset: 0

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Author

Paul D. Hanna, Apr 18 2007

Keywords

Examples

			Product formula is illustrated by:
A(x) = [1 + x + x^2 + x^3 + x^4 + x^5 +...]*
[1 + x^2 + 2x^3 + 3x^4 + 4x^5 + 5x^6 +...]*
[1 + x^3 + 3x^4 + 6x^5 + 10x^6 + 15x^7 +...]*
[1 + x^4 + 4x^5 + 10x^6 + 20x^7 + 35x^8 +...]*
[1 + x^5 + 5x^6 + 15x^7 + 35x^8 + 70x^9 +...]*...*
[1 + Sum_{k>=n+1} C(k-1,n)*x^k ]*...
		

Crossrefs

Programs

  • Mathematica
    Flatten[{1, Differences[Table[Sum[Binomial[n, k]*PartitionsQ[k], {k, 0, n}], {n, 0, 40}]]}] (* Vaclav Kotesovec, Oct 30 2017 *)
  • PARI
    {a(n)=polcoeff(prod(k=0,n,1+sum(i=k+1,n,binomial(i-1,k)*x^i +x*O(x^n))),n)}

Formula

G.f.: A(x) = Product_{n>=1} [1 + x^n/(1-x)^n].
a(n) = A266232(n) - A266232(n-1), for n>0. - Vaclav Kotesovec, Oct 30 2017
a(n) ~ exp(Pi*sqrt(n/6) + Pi^2/48) * 2^(n - 9/4) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 30 2017

A307500 Expansion of Product_{k>=1} 1/(1 - (x*(1 - x))^k).

Original entry on oeis.org

1, 1, 1, -1, -2, -4, 3, -1, 17, -16, 21, -57, 67, -130, 305, -536, 995, -1726, 2652, -4286, 7320, -13043, 24458, -45405, 81415, -141724, 239755, -400603, 676872, -1171076, 2072334, -3695550, 6519951, -11279015, 19188230, -32462795, 55334284, -95718737, 167673672, -294894076
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 11 2019

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 39; CoefficientList[Series[Product[1/(1 - (x (1 - x))^k), {k, 1, nmax}], {x, 0, nmax}], x]
    nmax = 39; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] (x (1 - x))^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
    Table[Sum[(-1)^(n - k) Binomial[k, n - k] PartitionsP[k], {k, 0, n}], {n, 0, 39}]

Formula

G.f.: exp(Sum_{k>=1} sigma(k)*(x*(1 - x))^k/k).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k,n-k)*A000041(k).

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).

A309575 Expansion of Product_{k>=1} (1 - (x*(1 + x))^k).

Original entry on oeis.org

1, -1, -2, -2, -1, 1, 5, 11, 17, 26, 36, 35, 20, -5, -65, -221, -510, -897, -1379, -2157, -3498, -5225, -6500, -6425, -4775, -1463, 5951, 25905, 74833, 173129, 334719, 563200, 876876, 1363232, 2208921, 3621969, 5631470, 7896109, 9725768, 10374574, 9340382, 6104500, -1413334
Offset: 0

Views

Author

Seiichi Manyama, Sep 21 2019

Keywords

Crossrefs

Convolution inverse of A238441.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1 - (x*(1+x))^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 22 2019 *)
  • 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).
Showing 1-4 of 4 results.