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 1-7 of 7 results.

A181541 E.g.f.: A(x) = Product_{n>=1} (1 + x^n/n)^n.

Original entry on oeis.org

1, 1, 2, 12, 54, 390, 3120, 28140, 290640, 3354960, 42561120, 586259520, 8806422240, 141680579040, 2446025662080, 44990666360640, 877867974023040, 18115179826423680, 394351821275892480, 9019730566889602560
Offset: 0

Views

Author

Paul D. Hanna, Nov 02 2010

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 12*x^3/3! + 54*x^4/4! + 390*x^5/5! + ...
A(x) = (1+x)*(1 + x^2/2)^2*(1 + x^3/3)^3*(1 + x^4/4)^4*(1 + x^5/5)^5*...

Links

Crossrefs

Cf. A007838, A294469.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1+x^k/k)^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 07 2020 *)
  • PARI
    {a(n)=n!*polcoeff(prod(m=1,n,(1+x^m/m+x*O(x^n))^m),n)}

Formula

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*(-j)^(k-1))). - Ilya Gutkovskiy, Sep 12 2018

A294470 E.g.f.: Product_{k>0} (1-x^k/k)^k.

Original entry on oeis.org

1, -1, -2, 0, 6, 90, 240, 2100, 1680, -28560, -682080, -7835520, -134497440, -1059458400, -13047914880, -138646347840, -1130671301760, 3558932657280, 93341485036800, 7903311136680960, 149238605943947520, 3997338754242113280, 86934078099901992960
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2017

Keywords

Links

Crossrefs

Cf. A292359.
Cf. A181541, A294469, A294471.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1-x^k/k)^k)))

A294471 E.g.f.: 1/Product_{k>0} (1+x^k/k)^k.

Original entry on oeis.org

1, -1, 0, -6, 18, -90, 660, -3360, 47880, -293160, 4277280, -36424080, 575190000, -6745218480, 101911249440, -1628086299840, 24861230634240, -484979925830400, 7629427896330240, -176975913961566720, 3036472694482106880, -77953392499390087680
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2017

Keywords

Links

Crossrefs

Cf. A292358.
Cf. A181541, A294469, A294470.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/Product[(1+x^k/k)^k,{k,nn}],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 30 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, (1+x^k/k)^k)))

A318693 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(i*j)/(i*j)).

Original entry on oeis.org

1, 1, 4, 16, 100, 628, 5388, 46212, 491328, 5381760, 68023056, 892073136, 13238778144, 201822014496, 3397195558560, 59356290115296, 1121097742183296, 21916440531679104, 459855848691876096, 9952944631606759680, 229191463614349301760, 5446997871156332605440, 136439919208493792455680
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 31 2018

Keywords

Crossrefs

Cf. A000005, A006171, A007841, A294469, A318415, A318694.

Programs

  • Maple
    seq(n!*coeff(series(mul(mul(1/(1-x^(i*j)/(i*j)),i=1..30),j=1..30),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Product[Product[1/(1 - x^(i j)/(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[1/(1 - x^k/k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[d^(1 - k/d) DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^(1 - k/d) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

Formula

E.g.f.: Product_{k>=1} 1/(1 - x^k/k)^tau(k), where tau = number of divisors (A000005).
E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} d^(1-k/d)*tau(d) ) * x^k/k).

A319176 a(n) = n! * [x^n] Product_{k>=1} 1/(1 - x^k/k)^n.

Original entry on oeis.org

1, 1, 8, 93, 1532, 32240, 829284, 25192454, 882825936, 35055329832, 1555548490560, 76285107738312, 4097094075364608, 239167754501235456, 15077741379436233120, 1020918130521930465120, 73892194568147257761024, 5693112248722998479169408, 465208700406183224884224000
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 12 2018

Keywords

Crossrefs

Cf. A007841, A294469, A299034, A319174, A319177.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Product[1/(1 - x^k/k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[Exp[n Sum[Sum[x^(j k)/(k j^k), {j, 1, n}], {k, 1, n}]], {x, 0, n}], {n, 0, 18}]

Formula

a(n) = n! * [x^n] exp(n*Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*j^k)).

A356560 Expansion of e.g.f. Product_{k>0} 1/(1 - k^2 * x^k)^(1/k^2).

Original entry on oeis.org

1, 1, 4, 18, 156, 1020, 16560, 143640, 2898000, 43016400, 926856000, 13749674400, 524416939200, 8626888670400, 284030505158400, 7950850859952000, 284397434953632000, 6752059834744224000, 357295791069689472000, 9098085523917918528000
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2022

Keywords

Crossrefs

Cf. A294462, A294469, A356530, A356561.
Cf. A077335, A308688.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-k^2*x^k)^(1/k^2))))
  • PARI
    a308688(n) = sumdiv(n, d, d^(2*n/d-1));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, a308688(j)*v[i-j+1]/(i-j)!)); v;

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A308688(k) * a(n-k)/(n-k)!.

A356561 Expansion of e.g.f. Product_{k>0} 1/(1 - k^3 * x^k)^(1/k^3).

Original entry on oeis.org

1, 1, 4, 18, 204, 1260, 37440, 299880, 11002320, 204860880, 6618628800, 92924647200, 8181137764800, 124123075876800, 7211104918617600, 288085376346768000, 14964000305173920000, 340302035937191328000, 42619767305209750656000
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2022

Keywords

Crossrefs

Cf. A294462, A294469, A356530, A356560.
Cf. A265837, A308689.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-k^3*x^k)^(1/k^3))))
  • PARI
    a308689(n) = sumdiv(n, d, d^(3*n/d-2));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, a308689(j)*v[i-j+1]/(i-j)!)); v;

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A308689(k) * a(n-k)/(n-k)!.
Showing 1-7 of 7 results.

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