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 1-8 of 8 results.

A294469 E.g.f.: 1/Product_{k>0} (1 - x^k/k)^k.

Original entry on oeis.org

1, 1, 4, 18, 114, 810, 7140, 68880, 766920, 9304680, 125086080, 1814015280, 28588356720, 481128888240, 8678237087520, 166041500264640, 3371031116893440, 72153115744469760, 1627441316510929920, 38500269726897538560, 954533425718494702080
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2017

Keywords

Links

Crossrefs

Cf. A007841, A181541, A294470, A294471.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(1-x^k/k)^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 01 2017 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, (1-x^k/k)^k)))

Formula

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

A182965 E.g.f.: A(x) = Product_{n>=1} (1 + 2*x^n/n)^n.

Original entry on oeis.org

1, 2, 4, 36, 168, 1440, 13920, 134400, 1619520, 20549760, 294631680, 4449096960, 74429752320, 1312794362880, 24870628823040, 501316411115520, 10661299747338240, 239672059847700480, 5664762159214878720
Offset: 0

Views

Author

Paul D. Hanna, Dec 19 2010

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x + 4*x^2/2! + 36*x^3/3! + 168*x^4/4! +...
A(x) = (1+2x)*(1+2x^2/2)^2*(1+2x^3/3)^3*(1+2x^4/4)^4*(1+2x^5/5)^5*...

Links

Crossrefs

Cf. A181541, A182966, A182967, A007838.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1 + 2*x^k/k)^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 07 2020 *)
  • PARI
    {a(n,k=2)=n!*polcoeff(prod(m=1,n,(1+k*x^m/m+x*O(x^n))^m),n)}

A182966 E.g.f.: A(x) = Product_{n>=1} (1 + 3*x^n/n)^n.

Original entry on oeis.org

1, 3, 6, 72, 342, 3330, 36720, 366660, 4974480, 67178160, 1043189280, 16836906240, 303306806880, 5705780240160, 114832957599360, 2475901844095680, 55754442891987840, 1331875774475326080, 33292197644365820160
Offset: 0

Views

Author

Paul D. Hanna, Dec 19 2010

Keywords

Examples

			E.g.f.: A(x) = 1 + 3*x + 6*x^2/2! + 72*x^3/3! + 342*x^4/4! +...
A(x) = (1+3x)*(1+3x^2/2)^2*(1+3x^3/3)^3*(1+3x^4/4)^4*(1+3x^5/5)^5*...

Links

Crossrefs

Cf. A181541, A182965, A182967, A007838.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1 + 3*x^k/k)^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 07 2020 *)
  • PARI
    {a(n,k=3)=n!*polcoeff(prod(m=1,n,(1+k*x^m/m+x*O(x^n))^m),n)}

A182967 E.g.f.: A(x) = Product_{n>=1} (1 + 4*x^n/n)^n.

Original entry on oeis.org

1, 4, 8, 120, 576, 6240, 75840, 772800, 11585280, 163914240, 2694558720, 45947489280, 876665180160, 17329568256000, 364677585592320, 8306018798837760, 195321474697789440, 4892032896606535680
Offset: 0

Views

Author

Paul D. Hanna, Dec 19 2010

Keywords

Examples

			E.g.f.: A(x) = 1 + 4*x + 8*x^2/2! + 120*x^3/3! + 576*x^4/4! +...
A(x) = (1+4x)*(1+4x^2/2)^2*(1+4x^3/3)^3*(1+4x^4/4)^4*(1+4x^5/5)^5*...

Links

Crossrefs

Cf. A181541, A182965, A182966, A007838.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Product[(1+4 x^n/n)^n,{n,nn}],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 11 2020 *)
  • PARI
    {a(n,k=4)=n!*polcoeff(prod(m=1,n,(1+k*x^m/m+x*O(x^n))^m),n)}

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)))

A318694 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(i*j)/(i*j)).

Original entry on oeis.org

1, 1, 2, 10, 40, 248, 1868, 14516, 131920, 1409040, 15697872, 191687472, 2663239104, 37878672960, 582357866400, 9898540886880, 172534018584960, 3192686545714560, 63844374067107840, 1309775114921541120, 28512040933544970240, 656888836504576112640, 15495311684125737031680
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 31 2018

Keywords

Links

Crossrefs

Cf. A000005, A007838, A107742, A181541, A318416, A318693.

Programs

  • Maple
    seq(n!*coeff(series(mul((1+x^k/k)^tau(k),k=1..100),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Product[Product[(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 + 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 + 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).

A319177 a(n) = n! * [x^n] Product_{k>=1} (1 + x^k/k)^n.

Original entry on oeis.org

1, 1, 4, 39, 500, 7990, 156684, 3640392, 97543088, 2960758800, 100428661440, 3764849536800, 154567280328768, 6897265807262064, 332386213584653760, 17204016957686536320, 951852354201532742400, 56059949872552858763520, 3501729575599545174352896, 231227806715994322631352960
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 12 2018

Keywords

Crossrefs

Cf. A007838, A181541, A300187, A319175, A319176.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Product[(1 + x^k/k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 19}]
Table[n! SeriesCoefficient[Exp[n Sum[Sum[(-1)^(k + 1) x^(j k)/(k j^k), {j, 1, n}], {k, 1, n}]], {x, 0, n}], {n, 0, 19}]

Formula

a(n) = n! * [x^n] exp(n*Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*j^k)).
Showing 1-8 of 8 results.

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