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 1-5 of 5 results.

A349583 E.g.f. satisfies: A(x) * log(A(x)) = exp(x) - 1.

Original entry on oeis.org

1, 1, 0, 2, -9, 72, -710, 8563, -121814, 1997502, -37097739, 769687954, -17644355410, 442894514285, -12081668234012, 355889274553166, -11258683640579857, 380701046875217492, -13702507978018209458, 523049811008797507683, -21105565578064063658754
Offset: 0

Views

Author

Seiichi Manyama, Nov 22 2021

Keywords

Crossrefs

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0,
         (1-m)^(m-1), m*b(n-1, m)+b(n-1, m+1))
        end:
    a:= n-> b(n, 0):
    seq(a(n), n=0..24);  # Alois P. Heinz, Jul 29 2022
  • Mathematica
    a[n_] := Sum[If[k == 1, 1, (-k + 1)^(k - 1)]*StirlingS2[n, k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Nov 23 2021 *)
  • PARI
    a(n) = sum(k=0, n, (-k+1)^(k-1)*stirling(n, k, 2));
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(exp(x)-1))))
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-k+1)^(k-1)*x^k/prod(j=1, k, 1-j*x)))

Formula

a(n) = Sum_{k=0..n} (-k+1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = exp( LambertW(exp(x) - 1) ).
G.f.: Sum_{k>=0} (-k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x).
a(n) ~ -(-1)^n * sqrt(exp(1) - 1) * n^(n-1) / (exp(n+1) * (1 - log(exp(1) - 1))^(n - 1/2)). - Vaclav Kotesovec, Dec 05 2021

A120980 E.g.f. satisfies: A(x)^A(x) = 1 + x.

Original entry on oeis.org

1, 1, -2, 9, -68, 740, -10554, 185906, -3891320, 94259952, -2592071760, 79748398752, -2713685928744, 101184283477680, -4102325527316184, 179674073609647080, -8454031849605513024, 425281651659459346944, -22777115050468598701248
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2006

Keywords

Crossrefs

Programs

  • Mathematica
    CoefficientList[Series[Log[1+x]/LambertW[Log[1+x]], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)
    Table[StirlingS1[n, 0] + StirlingS1[n, 1] + Sum[(-1)^(k + 1)*StirlingS1[n, k]*(k - 1)^(k - 1), {k, 2, n}], {n,0,50}] (* G. C. Greubel, Jun 21 2017 *)
    CoefficientList[Series[Exp[LambertW[Log[1+x]]], {x, 0, 25}], x]* Range[0, 25]! (* G. C. Greubel, Jun 22 2017 *)
  • PARI
    {a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,0); A[ #A]=-Vec(Ser(A)^Ser(A))[ #A]);n!*A[n+1]}
    
  • PARI
    x='x+O('x^50); Vec(serlaplace(exp(lambertw(log(1+x))))) \\ G. C. Greubel, Jun 22 2017

Formula

E.g.f.: A(x) = log(1+x)/LambertW(log(1+x)).
log(A(x)) = LambertW(log(1+x)).
E.g.f.: A(x) = 1/G(-x) where G(x) = g.f. of A052813.
E.g.f. of A052807 = -log(A(-x)) = -log(1-x)/A(-x).
a(n) = Sum_{k=0..n} (-1)^(k+1)*Stirling1(n,k)*(k-1)^(k-1). - Vladeta Jovovic, Jul 22 2006
|a(n)| ~ exp((exp(-1)-1)*n+3/2) * n^(n-1) / (exp(exp(-1))-1)^(n-1/2). - Vaclav Kotesovec, Jul 09 2013

A349656 E.g.f. satisfies: A(x)^2 * log(A(x)) = 1 - exp(-x).

Original entry on oeis.org

1, 1, -4, 35, -515, 10662, -284105, 9255185, -356346618, 15831168657, -797090201295, 44853942667096, -2789671436309939, 190023794141566309, -14069208182313480292, 1124994237749880216439, -96618656489949875115879, 8870165918232448251272870
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2021

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := (-1)^(n - 1) * Sum[(2*k - 1)^(k - 1)*StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)
  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, (2*k-1)^(k-1)*stirling(n, k, 2));
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(2*(1-exp(-x)))/2)))
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-2*k+1)^(k-1)*x^k/prod(j=1, k, 1+j*x)))

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} (2*k-1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = exp( LambertW(2*(1 - exp(-x)))/2 ).
G.f.: Sum_{k>=0} (-2*k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).
a(n) ~ -(-1)^n * sqrt(2*exp(1) + 1) * sqrt(-log(2) + log(2 + exp(-1))) * n^(n-1) / (2 * exp(n + 1/2) * (-log(2) + log(2*exp(1) + 1) - 1)^n). - Vaclav Kotesovec, Nov 24 2021

A349657 E.g.f. satisfies: A(x)^3 * log(A(x)) = 1 - exp(-x).

Original entry on oeis.org

1, 1, -6, 80, -1751, 53402, -2088528, 99680667, -5617170700, 365003288652, -26868393676609, 2209797209486528, -200828403704351068, 19986049281174575497, -2161617056877509895386, 252467067400866652634004, -31668302130310076212791823
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2021

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
    Do[A[x_] = Exp[(Exp[x]-1)/(Exp[x]*A[x]^3)]+O[x]^(nmax+1)//Normal, {nmax}];
    CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, (3*k-1)^(k-1)*stirling(n, k, 2));
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(3*(1-exp(-x)))/3)))
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-3*k+1)^(k-1)*x^k/prod(j=1, k, 1+j*x)))

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} (3*k-1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = exp( LambertW(3*(1 - exp(-x)))/3 ).
G.f.: Sum_{k>=0} (-3*k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).
a(n) ~ -(-1)^n * sqrt(3*exp(1) + 1) * sqrt(-log(3) + log(3 + exp(-1))) * n^(n-1) / (3 * exp(n + 1/3) * (-log(3) + log(3*exp(1) + 1) - 1)^n). - Vaclav Kotesovec, Nov 24 2021

A349602 E.g.f. satisfies: A(x) * log(A(x)) = 1 - exp(-x*A(x)^2).

Original entry on oeis.org

1, 1, 2, 2, -43, -668, -5908, -1209, 1399400, 37121106, 508366819, -3012861630, -444910083132, -15407930598279, -249403814792546, 5359691081465462, 589889204153846141, 23861630070579997032, 379819221897309026072, -21971010821241361939769
Offset: 0

Views

Author

Seiichi Manyama, Nov 22 2021

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(n - k)*(2*n - k + 1)^(k - 1)*StirlingS2[n, k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Nov 23 2021 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*(2*n-k+1)^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * (2*n-k+1)^(k-1) * Stirling2(n,k).
Showing 1-5 of 5 results.