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 1-10 of 10 results.

A356785 E.g.f. satisfies log(A(x)) = x * (exp(x*A(x)) - 1) * A(x).

Original entry on oeis.org

1, 0, 2, 3, 64, 365, 7356, 85687, 1920752, 34821369, 905128300, 22172123171, 672107454888, 20552960420005, 721088019634724, 26257726364294895, 1053711696230404576, 44336326818388565105, 2010106841636689325532, 95747319823049127621019
Offset: 0

Views

Author

Seiichi Manyama, Aug 27 2022

Keywords

Links

Crossrefs

Cf. A349560, A356788, A356789.
Cf. A184949, A349557, A355843.

Programs

  • Mathematica
    nmax = 19; A[_] = 1;
Do[A[x_] = Exp[x*(Exp[x*A[x]]-1)*A[x]] + 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\2, (n+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.
E.g.f.: (1/x) * Series_Reversion( x*exp(x*(1 - exp(x))) ). - Seiichi Manyama, Sep 21 2024

A356797 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x)^2.

Original entry on oeis.org

1, 0, 2, 3, 64, 305, 6936, 64897, 1645008, 24290289, 692240680, 14243244521, 456748635432, 12105737521033, 435619742434800, 14112089558682585, 567134312211275296, 21653262317886286817, 966207399513747354072, 42358800314758614030505
Offset: 0

Views

Author

Seiichi Manyama, Aug 28 2022

Keywords

Links

Crossrefs

Cf. A052506, A355843, A356798.
Cf. A356788.

Programs

  • Mathematica
    m = 20; (* number of terms *)
CoefficientList[Exp[-(1/2)*LambertW[-2*(Exp[x]-1)*x]] + O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, Sep 11 2022 *)
  • 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*(1-exp(x)))/2)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(2*x*(1-exp(x)))/(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 * (1 - exp(x)))/2 ).
E.g.f.: A(x) = ( LambertW(2 * x * (1 - exp(x)))/(2 * x * (1 - exp(x))) )^(1/2).

A357267 Expansion of e.g.f. -LambertW(x * (1 - exp(x))).

Original entry on oeis.org

0, 0, 2, 3, 28, 125, 1506, 12607, 186600, 2352681, 41839750, 705821171, 14818593516, 311784460429, 7603945309338, 190868446707135, 5328147004384336, 154893585657590609, 4884408906341245326, 161057122218190660555, 5671407469802947722900
Offset: 0

Views

Author

Seiichi Manyama, Sep 21 2022

Keywords

Links

Crossrefs

Cf. A048802, A355843, A357265.

Programs

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

Formula

a(n) = n! * Sum_{k=1..floor(n/2)} k^(k-1) * Stirling2(n-k,k)/(n-k)!.

A356798 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x)^3.

Original entry on oeis.org

1, 0, 2, 3, 88, 425, 13476, 130417, 4543120, 71005041, 2723297860, 60685651961, 2564091428856, 75166650583609, 3496499475113932, 127585829832674865, 6521845096842043936, 284745004488498858209, 15950013722559213419412, 809403234909367349670409
Offset: 0

Views

Author

Seiichi Manyama, Aug 28 2022

Keywords

Links

Crossrefs

Cf. A052506, A355843, A356797.
Cf. A356789.

Programs

  • Mathematica
    nmax = 19; A[_] = 1;
Do[A[x_] = Exp[x*(Exp[x] - 1)*A[x]^3] + 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, (3*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, (3*k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(3*x*(1-exp(x)))/3)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(3*x*(1-exp(x)))/(3*x*(1-exp(x))))^(1/3)))

Formula

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

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))).

A356902 E.g.f. satisfies A(x) * log(A(x)) = x * (exp(x) - 1).

Original entry on oeis.org

1, 0, 2, 3, -8, -55, 276, 4417, -13488, -639567, -248300, 141842921, 797525400, -43103642855, -584650622724, 16366430341185, 436555007091616, -6909610676492959, -368240758971238620, 2371795171252419385, 354876368637537736680, 1050192150132691993161
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A355843, A356797, A356798, A356904.
Cf. A349583.

Programs

  • Mathematica
    nmax = 21; A[_] = 1;
Do[A[x_] = Exp[(Exp[x] - 1)*x/A[x]] + 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\2, (-k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(x*(exp(x)-1)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-x*(1-exp(x))/lambertw(-x*(1-exp(x)))))

Formula

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

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))).

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).
Showing 1-10 of 10 results.

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