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.

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

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

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

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

A356199 a(n) = Sum_{k=0..n} (n*k+1)^(k-1) * Stirling2(n,k).

Original entry on oeis.org

1, 1, 6, 122, 5991, 556152, 84245291, 18956006323, 5940695613628, 2474958812797662, 1323229303771318595, 883245295259143164922, 719968321620942410875645, 703829776430361739799683993, 812798413118207226439408790038, 1094718407894086754989907938078190
Offset: 0

Views

Author

Alois P. Heinz, Jul 29 2022

Keywords

Links

Crossrefs

Cf. A000110, A008277, A048993, A052880, A349524, A349525, A349583, A349654, A349655.

Programs

  • Maple
    b:= proc(n, k, m) option remember; `if`(n=0,
     (k*m+1)^(m-1), m*b(n-1, k, m)+b(n-1, k, m+1))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..19);
  • Mathematica
    b[n_, k_, m_] := b[n, k, m] = If[n == 0,
   (k*m+1)^(m-1), m*b[n-1, k, m] + b[n-1, k, m+1]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Feb 14 2023, after Alois P. Heinz *)
  • PARI
    a(n) = sum(k=0, n, (n*k+1)^(k-1) * stirling(n, k, 2)); \\ Michel Marcus, Aug 04 2022

Formula

a(n) = Sum_{k=0..n} (n*k+1)^(k-1) * Stirling2(n,k).
a(n) = [x^n] Sum_{k>=0} (n*k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x).
a(n) = n! * [x^n] 1/exp(LambertW((1 - exp(x))*n)/n) for n > 0, a(0) = 1.
a(n) ~ exp(exp(-1)/2) * n^(2*n - 2). - Vaclav Kotesovec, Aug 07 2022

A363302 E.g.f. satisfies log(A(x)) = exp(x / A(x)^3) - 1.

Original entry on oeis.org

1, 1, -4, 41, -681, 15667, -460903, 16519141, -698242716, 34004778783, -1874858325725, 115438582354977, -7851013349413919, 584508287058281419, -47281383017104676456, 4129206143361098225405, -387216724567657721607901, 38806186875022459923785751
Offset: 0

Views

Author

Seiichi Manyama, May 26 2023

Keywords

Crossrefs

Cf. A030019, A334241, A349598, A349683, A362467.
Cf. A349655.

Programs

  • PARI
    a(n) = sum(k=0, n, (-3*n+1)^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (-3*n+1)^(k-1) * Stirling2(n,k).
Showing 1-6 of 6 results.

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