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.

A349654 E.g.f. satisfies: A(x)^2 * log(A(x)) = exp(x) - 1.

Original entry on oeis.org

1, 1, -2, 17, -213, 3712, -82773, 2250565, -72218912, 2671680015, -111950278213, 5240764049094, -271082407059027, 15353947287972373, -945097225235334538, 62820021683240176445, -4484426869618973019249, 342169496779859317566456
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2021

Keywords

Links

Crossrefs

Cf. A349650, A349652, A349656.
Cf. A008277, A349583, A349655.

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0,
     (1-2*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[(-2*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, (-2*k+1)^(k-1)*stirling(n, k, 2));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(2*(exp(x)-1))/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) = Sum_{k=0..n} (-2*k+1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = exp( LambertW(2*(exp(x) - 1))/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

A349650 E.g.f. satisfies: A(x)^(A(x)^2) = 1 + x.

Original entry on oeis.org

1, 1, -4, 36, -532, 11040, -295188, 9655772, -373422320, 16666348464, -843095987520, 47669276120928, -2979044176833888, 203906085094788960, -15170476121142482112, 1218972837861962011200, -105202043767190506428672, 9705514148732971389369600
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2021

Keywords

Links

Crossrefs

Cf. A349652, A349654, A349656.
Cf. A120980, A349651.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = (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)*abs(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) * exp(n*(exp(-1)/2 - 1)) / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n - 1/2)). - Vaclav Kotesovec, Nov 24 2021

A349653 E.g.f. satisfies: A(x)^(A(x)^3) = 1/(1 - x).

Original entry on oeis.org

1, 1, -4, 51, -996, 27120, -943602, 40023354, -2002953432, 115536775248, -7547711366880, 550798542893808, -44409102801760584, 3920444594317227600, -376109365694009875704, 38961901445878423746360, -4334496557343337848950208, 515407133679990302374396416
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2021

Keywords

Links

Crossrefs

Cf. A349651, A349655, A349657.
Cf. A349561, A349652.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = (1/(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)*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 * exp(1/6 - exp(-1)/6 - n) * n^(n-1) / (sqrt(3) * (-1 + exp(exp(-1)/3))^(n - 1/2)). - Vaclav Kotesovec, Nov 24 2021

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

Links

Crossrefs

Cf. A349650, A349652, A349654.
Cf. A008277, A349585, A349657.

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

A356906 E.g.f. satisfies A(x)^(A(x)^2) = 1/(1 - x)^x.

Original entry on oeis.org

1, 0, 2, 3, -28, -150, 2154, 26040, -322512, -7872984, 77570280, 3752301960, -22068935736, -2542757920560, 1422846762960, 2302464947491800, 14860063644794880, -2653728770258072640, -41790782141846648640, 3739260018343338345600
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Crossrefs

Cf. A355842, A356795, A356796, A356905.
Cf. A349652, A356885.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (-2*k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-2*k+1)^(k-1)*(-x*log(1-x))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-2*x*log(1-x))/2)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((-2*x*log(1-x)/lambertw(-2*x*log(1-x)))^(1/2)))

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (-2*k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (-2*k+1)^(k-1) * (-x * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-2 * x * log(1-x))/2 ).
E.g.f.: A(x) = ( -2 * x * log(1-x)/LambertW(-2 * x * log(1-x)) )^(1/2).
Showing 1-5 of 5 results.

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