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.

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A355781 E.g.f. satisfies log(A(x)) = 2 * (exp(x) - 1) * A(x).

Original entry on oeis.org

1, 2, 14, 166, 2854, 64854, 1839622, 62688406, 2497159302, 113932356630, 5860555367814, 335639363668118, 21184456464757894, 1461163816568091926, 109351697864286862214, 8825909581376322510230, 764231343305480319046278, 70670539764733828998689302
Offset: 0

Views

Author

Seiichi Manyama, Jul 16 2022

Keywords

Links

Crossrefs

Cf. A052880, A351276, A355788.

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0,
     2^m*(m+1)^(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
     b[n_, m_] := b[n, m] = If[n == 0, 2^m*(m + 1)^(m - 1), m*b[n - 1, m] + b[n - 1, m + 1]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 16 2022, after Alois P. Heinz *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*(1-exp(x))))))
  • PARI
    a(n) = sum(k=0, n, 2^k*(k+1)^(k-1)*stirling(n, k, 2));

Formula

E.g.f.: exp( -LambertW(2 * (1 - exp(x))) ).
a(n) = Sum_{k=0..n} 2^k * (k+1)^(k-1) * Stirling2(n,k).
From Vaclav Kotesovec, Jul 18 2022: (Start)
E.g.f.: LambertW(2 * (1 - exp(x))) / (2 * (1 - exp(x))).
a(n) ~ sqrt(2*exp(1) + 1) * sqrt(log(1 + exp(-1)/2)) * n^(n-1) / (exp(n-1) * (log(exp(1) + 1/2) - 1)^n). (End)

A356951 E.g.f. satisfies log(A(x)) = x^2/2 * (exp(x) - 1) * A(x).

Original entry on oeis.org

1, 0, 0, 3, 6, 10, 285, 1911, 8848, 155016, 1931625, 17006275, 276807036, 4801114968, 65672925409, 1172625764415, 24657199159440, 460156401399376, 9560083801337793, 230955040794126915, 5393971086379904260, 131545127670380245920, 3587507216606547324321
Offset: 0

Views

Author

Seiichi Manyama, Sep 06 2022

Keywords

Links

Crossrefs

Cf. A052880, A355843, A356952.
Cf. A000272, A354000, A356752, A356949.

Programs

  • Mathematica
    nmax = 22; A[_] = 1;
Do[A[x_] = Exp[x^2/2*(Exp[x] - 1)*A[x]] + 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\3, (k+1)^(k-1)*stirling(n-2*k, k, 2)/(2^k*(n-2*k)!));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x^2/2*(exp(x)-1))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^2/2*(1-exp(x))))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^2/2*(1-exp(x)))/(x^2/2*(1-exp(x)))))

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) * Stirling2(n-2*k,k)/(2^k * (n-2*k)!).
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x^2/2 * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( -LambertW(x^2/2 * (1 - exp(x))) ).
E.g.f.: A(x) = LambertW(x^2/2 * (1 - exp(x)))/(x^2/2 * (1 - exp(x))).

A356952 E.g.f. satisfies log(A(x)) = x^3/6 * (exp(x) - 1) * A(x).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 20, 35, 1736, 15204, 88320, 415965, 7632460, 121801966, 1368227224, 12184672955, 176889193040, 3490851044360, 59703361471296, 837948141904569, 13407228541467540, 283596013866706450, 6226093732482731800, 121326684752194084471
Offset: 0

Views

Author

Seiichi Manyama, Sep 06 2022

Keywords

Links

Crossrefs

Cf. A052880, A355843, A356951.
Cf. A000272, A354001, A356753, A356950.

Programs

  • Mathematica
    nmax = 23; A[_] = 1;
Do[A[x_] = Exp[x^3/6*(Exp[x] - 1)*A[x]] + 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\4, (k+1)^(k-1)*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x^3/6*(exp(x)-1))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^3/6*(1-exp(x))))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^3/6*(1-exp(x)))/(x^3/6*(1-exp(x)))))

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (k+1)^(k-1) * Stirling2(n-3*k,k)/(6^k * (n-3*k)!).
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x^3/6 * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( -LambertW(x^3/6 * (1 - exp(x))) ).
E.g.f.: A(x) = LambertW(x^3/6 * (1 - exp(x)))/(x^3/6 * (1 - exp(x))).

A356972 E.g.f. satisfies log(A(x)) = (exp(x * A(x)^2) - 1) * A(x).

Original entry on oeis.org

1, 1, 8, 128, 3139, 104382, 4393590, 224045271, 13428576766, 925335827928, 72082558060889, 6264277731652096, 600873473776204782, 63059026039778220285, 7187299097301622432156, 884141943373486896560252, 116756337165196381259759707, 16474480747756013055963484442
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2022

Keywords

Links

Crossrefs

Cf. A052880, A349557, A356973.
Cf. A216135, A349601, A356914.

Programs

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

Formula

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

A362799 E.g.f. satisfies A(x) = exp( (exp(x) - 1) * A(x)^x ).

Original entry on oeis.org

1, 1, 2, 11, 63, 542, 5183, 62211, 830252, 12900381, 220566835, 4223662522, 88001471869, 2007052809465, 49309469989666, 1306455781607975, 36973887007453315, 1116728635342926570, 35775769695237122035, 1213704083311914974899
Offset: 0

Views

Author

Seiichi Manyama, May 04 2023

Keywords

Links

Crossrefs

Cf. A052880, A362800.
Cf. A362794, A362796.
Cf. A000110, A361777.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*(exp(x)-1))/x)))

Formula

E.g.f.: exp( -LambertW(-x * (exp(x) - 1)) / x ).
E.g.f.: Sum_{k>=0} (k*x + 1)^(k-1) * (exp(x) - 1)^k / k!.

A362800 E.g.f. satisfies A(x) = exp( (exp(x) - 1) * A(x)^(x^2) ).

Original entry on oeis.org

1, 1, 2, 5, 39, 292, 2063, 21877, 271372, 3298155, 47855035, 805112970, 13843621861, 261388560253, 5529798475178, 122059754102345, 2863956966387107, 73150334575839340, 1961833778207602123, 55184622355007805281, 1656027290812446938492
Offset: 0

Views

Author

Seiichi Manyama, May 04 2023

Keywords

Links

Crossrefs

Cf. A052880, A362799.
Cf. A362795, A362798.
Cf. A000110, A362571.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x^2*(exp(x)-1))/x^2)))

Formula

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

A356949 E.g.f. satisfies log(A(x)) = x^2 * (exp(x) - 1) * A(x).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 1110, 7602, 35336, 1103832, 14984010, 134552990, 3457329612, 70828191876, 1017237973934, 25648737955050, 676111332667920, 13760810592066992, 373071111301807506, 11594147432172228918, 307097278689726728660, 9330499711181779575900
Offset: 0

Views

Author

Seiichi Manyama, Sep 06 2022

Keywords

Links

Crossrefs

Cf. A052880, A355843, A356950.
Cf. A000272, A240989, A355287, A356797, A356951.

Programs

  • Mathematica
    nmax = 21; A[_] = 1;
Do[A[x_] = Exp[(-1 + Exp[x])*A[x]*x^2] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\3, (k+1)^(k-1)*stirling(n-2*k, k, 2)/(n-2*k)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x^2*(exp(x)-1))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^2*(1-exp(x))))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^2*(1-exp(x)))/(x^2*(1-exp(x)))))

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) * Stirling2(n-2*k,k)/(n-2*k)!.
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x^2 * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( -LambertW(x^2 * (1 - exp(x))) ).
E.g.f.: A(x) = LambertW(x^2 * (1 - exp(x)))/(x^2 * (1 - exp(x))).

A356950 E.g.f. satisfies log(A(x)) = x^3 * (exp(x) - 1) * A(x).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 120, 210, 60816, 544824, 3175920, 14969790, 1339209960, 25141598196, 291418089144, 2618105492730, 128974591028640, 3841451570440560, 73103023032142176, 1060951475511351414, 39132892925113341240, 1516348247446904304300
Offset: 0

Views

Author

Seiichi Manyama, Sep 06 2022

Keywords

Links

Crossrefs

Cf. A052880, A355843, A356949.
Cf. A000272, A356099, A356952.

Programs

  • Mathematica
    nmax = 21; A[_] = 1;
Do[A[x_] = Exp[(-1 + Exp[x])*A[x]*x^3] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\4, (k+1)^(k-1)*stirling(n-3*k, k, 2)/(n-3*k)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x^3*(exp(x)-1))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^3*(1-exp(x))))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^3*(1-exp(x)))/(x^3*(1-exp(x)))))

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (k+1)^(k-1) * Stirling2(n-3*k,k)/(n-3*k)!.
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x^3 * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( -LambertW(x^3 * (1 - exp(x))) ).
E.g.f.: A(x) = LambertW(x^3 * (1 - exp(x)))/(x^3 * (1 - exp(x))).

A357009 E.g.f. satisfies log(A(x)) = (exp(x) - 1)^2 * A(x).

Original entry on oeis.org

1, 0, 2, 6, 50, 390, 4322, 53046, 782210, 12920550, 241747682, 5000171286, 113961184130, 2830240421190, 76196913418082, 2209152734071926, 68655746019566210, 2276606079902438310, 80244521295497399522, 2995966456305973559766, 118119901491333724203650
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2022

Keywords

Links

Crossrefs

Cf. A052880, A357010.
Cf. A052859, A357024.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(-1 + Exp[x])^2*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = sum(k=0, n\2, (2*k)!*(k+1)^(k-1)*stirling(n, 2*k, 2)/k!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(exp(x)-1)^(2*k)/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-(exp(x)-1)^2))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-lambertw(-(exp(x)-1)^2)/(exp(x)-1)^2))

Formula

a(n) = Sum_{k=0..floor(n/2)} (2*k)! * (k+1)^(k-1) * Stirling2(n,2*k)/k!.
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (exp(x) - 1)^(2*k) / k!.
E.g.f.: A(x) = exp( -LambertW(-(exp(x) - 1)^2) ).
E.g.f.: A(x) = -LambertW(-(exp(x) - 1)^2)/(exp(x) - 1)^2.
a(n) ~ sqrt(1 + exp(1/2)) * 2^n * n^(n-1) / (exp(n-1) * (2*log(1 + exp(1/2)) - 1)^(n - 1/2)). - Vaclav Kotesovec, Sep 27 2023

A357010 E.g.f. satisfies log(A(x)) = (exp(x) - 1)^3 * A(x).

Original entry on oeis.org

1, 0, 0, 6, 36, 150, 1620, 24486, 293076, 3843510, 68254740, 1311687366, 25479935316, 552545882070, 13437670215060, 345157499363046, 9370414233900756, 274413997443811830, 8572526271218671380, 281754864204797848326, 9767868351458229261396
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2022

Keywords

Links

Crossrefs

Cf. A052880, A357009.
Cf. A353664, A357025.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(Exp[x] - 1)^3*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*(k+1)^(k-1)*stirling(n, 3*k, 2)/k!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(exp(x)-1)^(3*k)/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-(exp(x)-1)^3))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-lambertw(-(exp(x)-1)^3)/(exp(x)-1)^3))

Formula

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * (k+1)^(k-1) * Stirling2(n,3*k)/k!.
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (exp(x) - 1)^(3*k) / k!.
E.g.f.: A(x) = exp( -LambertW(-(exp(x) - 1)^3) ).
E.g.f.: A(x) = -LambertW(-(exp(x) - 1)^3)/(exp(x) - 1)^3.
a(n) ~ sqrt(1 + exp(1/3)) * 3^n * n^(n-1) / (exp(n-1) * (3*log(1 + exp(1/3)) - 1)^(n - 1/2)). - Vaclav Kotesovec, Sep 27 2023
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