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.

A349655 E.g.f. satisfies: A(x)^3 * log(A(x)) = exp(x) - 1.

Original entry on oeis.org

1, 1, -4, 50, -981, 26632, -924526, 39114343, -1952373450, 112321286934, -7318049389727, 532602419776770, -42825957593127770, 3770431528821292441, -360734325565272740984, 37267364164988863692782, -4134667018838875759388749, 490302545213321842575157140
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2021

Keywords

Links

Crossrefs

Cf. A349651, A349653, A349657.
Cf. A008277, A349583, A349654.

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0,
     (1-3*m)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..21);  # Alois P. Heinz, Jul 29 2022
  • Mathematica
    a[n_] := Sum[(-3*k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)
  • PARI
    a(n) = 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*(exp(x)-1))/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) = Sum_{k=0..n} (-3*k+1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = exp( LambertW(3*(exp(x) - 1))/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

A349652 E.g.f. satisfies: A(x)^(A(x)^2) = 1/(1 - x).

Original entry on oeis.org

1, 1, -2, 18, -220, 3880, -86868, 2376836, -76735216, 2856604464, -120457684320, 5675047644288, -295430737430112, 16840861797433440, -1043322313406139648, 69798144929293516800, -5014888682767294232832, 385130588783629323238656
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2021

Keywords

Links

Crossrefs

Cf. A349650, A349654, A349656.
Cf. A349561, A349653.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = (1/(1 - x))^(1/A[x]^2) + 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, (2*k-1)^(k-1)*stirling(n, k, 1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (2*k-1)^(k-1)*log(1-x)^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((-2*log(1-x)/lambertw(-2*log(1-x)))^(1/2)))

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} (2*k-1)^(k-1) * Stirling1(n,k).
E.g.f. A(x) = -Sum_{k>=0} (2*k-1)^(k-1) * (log(1-x))^k / k!.
E.g.f.: A(x) = ( -2*log(1-x)/LambertW(-2*log(1-x)) )^(1/2).
a(n) ~ -(-1)^n * n^(n-1) / (sqrt(2) * exp(n + exp(-1)/4) * (-1 + exp(exp(-1)/2))^(n - 1/2)). - Vaclav Kotesovec, Nov 24 2021

A349651 E.g.f. satisfies: A(x)^(A(x)^3) = 1 + x.

Original entry on oeis.org

1, 1, -6, 81, -1776, 54240, -2125122, 101631558, -5739235128, 373745355984, -27572590788480, 2272763834553168, -207013811669644680, 20647997125333476912, -2238256520486195804280, 262010379635788799196360, -32939968662220720559744448
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2021

Keywords

Links

Crossrefs

Cf. A349653, A349655, A349657.
Cf. A120980, A349650.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = (1 + x)^(1/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)*abs(stirling(n, k, 1)));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (3*k-1)^(k-1)*(-log(1+x))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((3*log(1+x)/lambertw(3*log(1+x)))^(1/3)))

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} (3*k-1)^(k-1) * |Stirling1(n,k)|.
E.g.f. A(x) = -Sum_{k>=0} (3*k-1)^(k-1) * (-log(1+x))^k / k!.
E.g.f.: A(x) = ( 3*log(1+x)/LambertW(3*log(1+x)) )^(1/3).
a(n) ~ -(-1)^n * n^(n-1) * exp(1/6 - n + n*exp(-1)/3) / (sqrt(3) * (exp(exp(-1)/3) - 1)^(n - 1/2)). - 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

Links

Crossrefs

Cf. A349651, A349653, A349655.
Cf. A349585, A349656.

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
Showing 1-4 of 4 results.

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