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 1-8 of 8 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

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

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

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

A355763 E.g.f. satisfies A(x)^2 * log(A(x)) = exp(x*A(x)) - 1.

Original entry on oeis.org

1, 1, 0, 5, -13, 207, -1791, 28849, -438600, 8619291, -181134313, 4381744589, -115439041983, 3356162869607, -105668550658100, 3600058076291465, -131618721053773713, 5146452228945999699, -214171122214841864975, 9454288479242533668837
Offset: 0

Views

Author

Seiichi Manyama, Jul 16 2022

Keywords

Crossrefs

Cf. A030019, A349588.
Cf. A349654, A355762.

Programs

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

Formula

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

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

A356904 E.g.f. satisfies A(x)^2 * log(A(x)) = x * (exp(x) - 1).

Original entry on oeis.org

1, 0, 2, 3, -32, -175, 2376, 29617, -371440, -9251919, 91421560, 4529155961, -26677647864, -3160004989271, 1541460644192, 2946529440977865, 19556193589426336, -3498019439220155551, -56274505323609293208, 5077223330715030358009
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A355843, A356797, A356798, A356902.
Cf. A349654.

Programs

  • Mathematica
    nmax = 19; A[_] = 1;
Do[A[x_] = Exp[((Exp[x]-1)*x)/A[x]^2] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\2, (-2*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-2*k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(2*x*(exp(x)-1))/2)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((-2*x*(1-exp(x))/lambertw(-2*x*(1-exp(x))))^(1/2)))

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (-2*k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (-2*k+1)^(k-1) * (x * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( LambertW(2 * x * (exp(x) - 1))/2 ).
E.g.f.: A(x) = ( -2 * x * (1 - exp(x))/LambertW(-2 * x * (1 - exp(x))) )^(1/2).

A362467 E.g.f. satisfies log(A(x)) = exp(x / A(x)^2) - 1.

Original entry on oeis.org

1, 1, -2, 11, -97, 1162, -17401, 309297, -6284804, 141430821, -3411964025, 84469913200, -1971020234987, 31982106694501, 703663251081166, -131978337454653865, 11571772746664732291, -879307513026396837470, 64266964230133042267891
Offset: 0

Views

Author

Seiichi Manyama, May 26 2023

Keywords

Crossrefs

Cf. A030019, A334241, A349598, A349683, A363302.
Cf. A349654.

Programs

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

Formula

a(n) = Sum_{k=0..n} (-2*n+1)^(k-1) * Stirling2(n,k).
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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