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.

A052807 Expansion of e.g.f. -LambertW(log(1-x)).

Original entry on oeis.org

0, 1, 3, 17, 146, 1704, 25284, 456224, 9702776, 237711888, 6593032560, 204212077992, 6986942528400, 261700394006232, 10650713784774504, 468007296229553880, 22083086552247101184, 1113646609708909274880
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

Previous name was: A simple grammar.
E.g.f. of A052813 equals exp(A(x)) = -A(x)/log(1-x). a(n) = n!*Sum_{k=0..n-1} A052813(k)/k!/(n-k). - Paul D. Hanna, Jul 19 2006

Examples

			E.g.f.: A(x) = x + 3*x^2/2! + 17*x^3/3! + 146*x^4/4! +...
A(x)/exp(A(x)) = -log(1-x) = x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 +...

Links

Crossrefs

Cf. A006963, A048802, A052813 (exp(A(x))), A277489.

Programs

  • Maple
    spec := [S,{B=Cycle(Z),C=Set(S),S=Prod(C,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    max = 17; se = Series[-ProductLog[-Log[-(-1 + x)^(-1)]] , {x, 0, max}]; Join[{0}, (CoefficientList[se, x] // DeleteCases[#, 0] &) * Range[max]!] (* Jean-François Alcover, Jun 24 2013 *)
CoefficientList[Series[-LambertW[-Log[-1/(-1 + x)]], {x,0,50}], x]*
Range[0,50]! (* G. C. Greubel, Jun 18 2017 *)
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1/(1-x+x*O(x^n))^A);n!*polcoeff(log(A),n)} \\ Paul D. Hanna, Jul 19 2006
  • PARI
    x = 'x + O('x^30); concat(0, Vec(serlaplace(-lambertw(log(1-x))))) \\ Michel Marcus, Jun 19 2017

Formula

a(n) = Sum_{k=1..n} |Stirling1(n, k)|*k^(k-1). - Vladeta Jovovic, Sep 17 2003
E.g.f. satisfies: A(x) = 1/(1-x)^A(x). - Paul D. Hanna, Jul 19 2006
a(n) ~ n^(n-1)*exp((exp(-1)-1)*n+1/2) / (exp(exp(-1))-1)^(n-1/2). - Vaclav Kotesovec, Jul 09 2013
E.g.f.: Series_Reversion( 1 - exp(-x*exp(-x)) ). - Seiichi Manyama, Sep 08 2024

Extensions

New name using e.g.f. by Vaclav Kotesovec, Oct 18 2013

A305819 Expansion of e.g.f. 1/(1 + LambertW(-log(1 + x))).

Original entry on oeis.org

1, 1, 3, 17, 132, 1334, 16442, 239994, 4041776, 77183328, 1647541632, 38877352392, 1004869488048, 28234217634024, 856830099396840, 27930093941047464, 973269467390922240, 36104568839480990400, 1420556927968241858880, 59088101641333114906944, 2590680379402887359111424
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 18 2018

Keywords

Comments

Inverse Stirling transform of A000312.

Links

Crossrefs

Cf. A000312, A052807, A277489, A282190, A305981.

Programs

  • Maple
    a:=series(1/(1+LambertW(-log(1+x))),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 + LambertW[-Log[1 + x]]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] k^k, {k, n}], {n, 20}]]
  • PARI
    a(n) = sum(k=0, n, k^k*stirling(n, k, 1)); \\ Seiichi Manyama, Feb 05 2022
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-log(1+x))))) \\ Seiichi Manyama, Feb 05 2022

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*k^k.
a(n) ~ n^n / ((exp(exp(-1)) - 1)^(n + 1/2) * exp(n + (1 - exp(-1))/2)). - Vaclav Kotesovec, Aug 18 2018

A305981 Expansion of e.g.f. 1/(1 + LambertW(log(1 - x))).

Original entry on oeis.org

1, 1, 5, 41, 468, 6854, 122582, 2589978, 63129392, 1743732192, 53827681152, 1836453542472, 68620052332752, 2786929842106344, 122241516227220504, 5758920745460806824, 290017142065771138560, 15547326972257789803200, 883974436758296523437760, 53131928820278417749940544, 3366145488853852112016117504
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 18 2018

Keywords

Links

Crossrefs

Cf. A000312, A052807, A277489, A282190, A305819.

Programs

  • Maple
    a:=series(1/(1+LambertW(log(1-x))),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 + LambertW[Log[1 - x]]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[Abs[StirlingS1[n, k]] k^k, {k, n}], {n, 20}]]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*k^k*stirling(n, k, 1)); \\ Seiichi Manyama, Feb 05 2022
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(log(1-x))))) \\ Seiichi Manyama, Feb 05 2022

Formula

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*k^k.
a(n) ~ n^n / ((exp(exp(-1)) - 1)^(n + 1/2) * exp(n*(1 - exp(-1)) + 1/2)). - Vaclav Kotesovec, Aug 18 2018

A357338 E.g.f. satisfies A(x) = log(1 + x) * exp(3 * A(x)).

Original entry on oeis.org

0, 1, 5, 65, 1302, 35904, 1260372, 53796168, 2704942440, 156602951568, 10260496538640, 750563024381928, 60636437884772208, 5362045857366832152, 515154874732515894744, 53432840588453561773080, 5950904875941534263739648, 708296073287989866587094528
Offset: 0

Views

Author

Seiichi Manyama, Sep 24 2022

Keywords

Links

Crossrefs

Cf. A277489, A357337.
Cf. A349505.

Programs

  • Mathematica
    nmax = 20; A[_] = 0;
Do[A[x_] = Log[1 + x]*Exp[3*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)
  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-3*log(1+x))/3)))
  • PARI
    a(n) = sum(k=1, n, (3*k)^(k-1)*stirling(n, k, 1));

Formula

E.g.f.: -LambertW(-3 * log(1 + x))/3.
a(n) = Sum_{k=1..n} (3 * k)^(k-1) * Stirling1(n,k).
a(n) ~ n^(n-1) / (sqrt(3) * (exp(exp(-1)/3) - 1)^(n - 1/2) * exp(n - 1/2 + exp(-1)/6)). - Vaclav Kotesovec, Oct 04 2022
E.g.f.: Series_Reversion( exp(x * exp(-3*x)) - 1 ). - Seiichi Manyama, Sep 10 2024

A357337 E.g.f. satisfies A(x) = log(1 + x) * exp(2 * A(x)).

Original entry on oeis.org

0, 1, 3, 26, 334, 5964, 135228, 3729872, 121172560, 4532603904, 191869653120, 9067948437888, 473297792213376, 27039987154142208, 1678363256057198592, 112467293224249912320, 8092242817059530284032, 622253112192770288799744
Offset: 0

Views

Author

Seiichi Manyama, Sep 24 2022

Keywords

Links

Crossrefs

Cf. A277489, A357338.
Cf. A349504.

Programs

  • Mathematica
    nmax = 20; A[_] = 0;
Do[A[x_] = Log[1 + x]*Exp[2*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)
  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*log(1+x))/2)))
  • PARI
    a(n) = sum(k=1, n, (2*k)^(k-1)*stirling(n, k, 1));

Formula

E.g.f.: -LambertW(-2 * log(1 + x))/2.
a(n) = Sum_{k=1..n} (2 * k)^(k-1) * Stirling1(n,k).
a(n) ~ n^(n-1) / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n - 1/2) * exp(n + exp(-1)/4 - 1/2)). - Vaclav Kotesovec, Oct 04 2022
E.g.f.: Series_Reversion( exp(x * exp(-2*x)) - 1 ). - Seiichi Manyama, Sep 10 2024

A323619 Expansion of e.g.f. 1 - LambertW(-log(1+x))*(2 + LambertW(-log(1+x)))/2.

Original entry on oeis.org

1, 1, 0, 2, 3, 44, 260, 3534, 40796, 658440, 11066184, 220005840, 4750650432, 114430365048, 2993377996440, 85208541290040, 2611784941760640, 85941161628865344, 3018822193183216320, 112805065528683216192, 4467115744449046110720, 186900232401341222964480, 8237944325702047624948224
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 20 2019

Keywords

Links

Crossrefs

Cf. A000272, A038052, A048994, A277489, A305819.

Programs

  • Magma
    [n le 0 select 1 else (&+[StirlingFirst(n,k)*k^(k-2): k in [1..n]]): n in [0..25]]; // G. C. Greubel, Feb 07 2019
  • Maple
    seq(n!*coeff(series(1-LambertW(-log(1+x))*(2+LambertW(-log(1+x)))/2, x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 28 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[1 - LambertW[-Log[1 + x]] (2 + LambertW[-Log[1 + x]])/2, {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] k^(k - 2), {k, n}], {n, 22}]]
  • PARI
    {a(n) = if(n==0,1, sum(k=1,n, stirling(n,k,1)*k^(k-2)))};
vector(25, n, n--; a(n)) \\ G. C. Greubel, Feb 07 2019
  • Sage
    [1] + [sum((-1)^(k+n)*stirling_number1(n,k)*k^(k-2) for k in (1..n)) for n in (1..25)] # G. C. Greubel, Feb 07 2019

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000272(k).
a(n) ~ n^(n-2) / ((exp(exp(-1))-1)^(n - 3/2) * exp(n - 3*(1 - exp(-1))/2)). - Vaclav Kotesovec, Jan 20 2019
Showing 1-6 of 6 results.

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