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.

Previous Showing 11-19 of 19 results.

A362655 E.g.f. satisfies A(x) = exp( x * exp(x^3) * A(x) ).

Original entry on oeis.org

1, 1, 3, 16, 149, 1656, 22567, 369664, 7081209, 155178928, 3830958251, 105267080304, 3187172910517, 105437661606616, 3784329536385231, 146474021771040856, 6081955388047685873, 269686446704697314016, 12719466142269818201299
Offset: 0

Views

Author

Seiichi Manyama, Apr 28 2023

Keywords

Links

Crossrefs

Cf. A273954, A362654.
Cf. A354553, A362674.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*exp(x^3)))))

Formula

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

A362671 E.g.f. satisfies A(x) = exp( x * exp(x) / A(x)^2 ).

Original entry on oeis.org

1, 1, -1, 10, -111, 1716, -33755, 807738, -22782207, 740204776, -27226430739, 1118416240470, -50750734988063, 2521219487859372, -136098630522431499, 7932551567421395866, -496501182232557828735, 33214032504796887027408
Offset: 0

Views

Author

Seiichi Manyama, Apr 29 2023

Keywords

Links

Crossrefs

Cf. A273954, A360473, A362656, A362672.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(2*x*exp(x))/2)))

Formula

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

A362672 E.g.f. satisfies A(x) = exp( x * exp(x) / A(x)^3 ).

Original entry on oeis.org

1, 1, -3, 37, -679, 17161, -553451, 21731053, -1006118863, 53671172113, -3241671266899, 218677223408821, -16296163119155063, 1329568681331536153, -117874745761237043515, 11283758432396431997821, -1159952212029532257663391, 127445385808282289840496673
Offset: 0

Views

Author

Seiichi Manyama, Apr 29 2023

Keywords

Links

Crossrefs

Cf. A273954, A360473, A362656, A362671.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(3*x*exp(x))/3)))

Formula

E.g.f.: exp( LambertW(3*x * exp(x))/3 ).
a(n) = Sum_{k=0..n} k^(n-k) * (-3*k+1)^(k-1) * binomial(n,k).

A380406 E.g.f. satisfies A(x) = exp( 2 * x * exp(x) * A(x)^(1/2) ).

Original entry on oeis.org

1, 2, 12, 104, 1232, 18592, 342208, 7451264, 187631872, 5369721344, 172255038464, 6125052946432, 239195824279552, 10179739052908544, 469024768235192320, 23263095316577681408, 1235978286454556131328, 70040404736026578386944, 4217180561907991530176512
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2025

Keywords

Links

Crossrefs

Cf. A273954, A357247, A380407.
Cf. A273953, A360473.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A273954.
E.g.f.: A(x) = exp( -2*LambertW(-x * exp(x)) ).
a(n) = 2 * Sum_{k=0..n} k^(n-k) * (k+2)^(k-1) * binomial(n,k).
a(n) ~ 2 * sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (exp(n-2) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Aug 05 2025

A380407 E.g.f. satisfies A(x) = exp( 3 * x * exp(x) * A(x)^(1/3) ).

Original entry on oeis.org

1, 3, 21, 207, 2697, 43803, 854685, 19512615, 510977937, 15112457523, 498560461989, 18160560320895, 724240913035545, 31394996915447883, 1470245245400432685, 73987438021589516247, 3982389565847576723745, 228331703268783136636515, 13894569264190369648271157
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2025

Keywords

Links

Crossrefs

Cf. A273954, A357247, A380406.

Programs

  • Mathematica
    terms = 19; A[] = 0; Do[A[x] = Exp[3*x*Exp[x]*A[x]^(1/3)] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Jun 14 2025 *)
  • PARI
    a(n) = 3*sum(k=0, n, k^(n-k)*(k+3)^(k-1)*binomial(n, k));

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A273954.
E.g.f.: A(x) = exp( -3*LambertW(-x * exp(x)) ).
a(n) = 3 * Sum_{k=0..n} k^(n-k) * (k+3)^(k-1) * binomial(n,k).

A360466 E.g.f. satisfies A(x) = exp(2 * x * exp(x) * A(x)).

Original entry on oeis.org

1, 2, 16, 206, 3832, 93962, 2871820, 105355406, 4515648784, 221598121490, 12257187851284, 754703476252310, 51204818674338328, 3796079000648275226, 305328667748448560668, 26483633169003911205278, 2464307301750079915255840, 244872778601760932275686434
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2023

Keywords

Links

Crossrefs

Cf. A273954, A357247, A360465.

Programs

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

Formula

E.g.f.: A(x) = exp( -LambertW(-2 * x * exp(x)) ).
E.g.f.: A(x) = -LambertW(-2 * x * exp(x)) / (2 * x * exp(x)).
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (2 * x * exp(x))^k / k!.
a(n) = Sum_{k=0..n} 2^k * k^(n-k) * (k+1)^(k-1) * binomial(n,k).
a(n) ~ sqrt(1+LambertW(exp(-1)/2)) * n^(n-1) / (exp(n-1) * LambertW(exp(-1)/2)^n). - Vaclav Kotesovec, Feb 08 2023

A362660 E.g.f. satisfies A(x) = exp( x * exp(x^2/2) * A(x) ).

Original entry on oeis.org

1, 1, 3, 19, 161, 1791, 24847, 413449, 8036625, 178852753, 4486426091, 125279093259, 3854964555697, 129618443364463, 4728625129171959, 186034319795094481, 7851808690935373793, 353903271319498588641, 16966669198377512202643
Offset: 0

Views

Author

Seiichi Manyama, Apr 29 2023

Keywords

Links

Crossrefs

Cf. A273954, A362661.
Cf. A354550, A362654.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*exp(x^2/2)))))

Formula

E.g.f.: exp( -LambertW(-x * exp(x^2/2)) ).
a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)^k * (n-2*k+1)^(n-2*k-1) / (2^k * k! * (n-2*k)!).
a(n) ~ sqrt(1 + LambertW(exp(-2))) * n^(n-1) / (exp(n-1) * LambertW(exp(-2))^(n/2)). - Vaclav Kotesovec, Aug 05 2025

A362661 E.g.f. satisfies A(x) = exp( x * exp(x^3/6) * A(x) ).

Original entry on oeis.org

1, 1, 3, 16, 129, 1356, 17767, 279714, 5149209, 108591688, 2582351451, 68380940904, 1995777685717, 63659665732716, 2203395556479951, 82253291389678756, 3294326092613575473, 140911264444599281616, 6411278790217738946899
Offset: 0

Views

Author

Seiichi Manyama, Apr 29 2023

Keywords

Links

Crossrefs

Cf. A273954, A362660.
Cf. A143567, A354551, A362655.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*exp(x^3/6)))))

Formula

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

A372279 E.g.f. A(x) satisfies A(x) = exp( x * ( exp(x) * A(x) )^(5/2) ).

Original entry on oeis.org

1, 1, 11, 181, 4461, 148101, 6202651, 314158461, 18682884681, 1276509416761, 98552772971451, 8485633118339301, 806247602665104661, 83796784405535693181, 9457590223483413296811, 1151924494605809502276301, 150602291336042725831941201
Offset: 0

Views

Author

Seiichi Manyama, Apr 25 2024

Keywords

Links

Crossrefs

Cf. A273953, A273954, A360544, A360547.
Cf. A372278.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2/5*lambertw(-5/2*x*exp(5*x/2)))))
  • PARI
    a(n) = sum(k=0, n, (5*k/2)^(n-k)*(5*k/2+1)^(k-1)*binomial(n, k));

Formula

E.g.f.: A(x) = exp( -2/5 * LambertW(-5*x/2 * exp(5*x/2)) ).
E.g.f.: A(x) = ( -LambertW(-5*x/2 * exp(5*x/2)) / (5*x/2 * exp(5*x/2)) )^(2/5).
E.g.f.: A(x) = ( Sum_{k>=0} (k+1)^(k-1) * (5*x/2 * exp(5*x/2))^k / k! )^(2/5).
a(n) = Sum_{k=0..n} (5*k/2)^(n-k) * (5*k/2+1)^(k-1) * binomial(n,k).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 5^(n-1) * n^(n-1) / (exp(n - 2/5) * 2^(n-1) * LambertW(exp(-1))^n). - Vaclav Kotesovec, May 06 2024
Previous Showing 11-19 of 19 results.

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