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 1-5 of 5 results.

A362694 E.g.f. satisfies A(x) = exp(x + x * A(x)^2).

Original entry on oeis.org

1, 2, 12, 152, 2960, 78112, 2607808, 105432448, 5008584960, 273482293760, 16878251101184, 1161918967060480, 88277165100666880, 7337286679766179840, 662287143981044121600, 64516370031367063175168, 6746443728505612426870784, 753763691778003738319519744
Offset: 0

Views

Author

Seiichi Manyama, May 01 2023

Keywords

Links

Crossrefs

Cf. A349562, A362693, A362734, A362735.
Cf. A143768, A202617.

Programs

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

Formula

E.g.f.: sqrt( -LambertW(-2*x*exp(2*x)) / (2*x) ) = exp( x - LambertW(-2*x*exp(2*x))/2 ).
a(n) = Sum_{k=0..n} (2*k+1)^(n-1) * binomial(n,k) = 2^n * A202617(n).
a(n) ~ sqrt(1 + 1/LambertW(exp(-1))) * 2^(n-1) * n^(n-1) / (exp(n) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Nov 10 2023

A362734 E.g.f. satisfies A(x) = exp(x + x * A(x)^3).

Original entry on oeis.org

1, 2, 16, 296, 8512, 333632, 16595200, 1001460224, 71094759424, 5805799829504, 536188352856064, 55259197654089728, 6287146625230962688, 782751635353947865088, 105852868748672770244608, 15451195442132410179780608, 2421355190097788960505856000
Offset: 0

Views

Author

Seiichi Manyama, May 01 2023

Keywords

Links

Crossrefs

Cf. A349562, A362693, A362694, A362735.
Cf. A349714, A362392, A362472.

Programs

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

Formula

E.g.f.: ( -LambertW(-3*x*exp(3*x)) / (3*x) )^(1/3) = exp( x - LambertW(-3*x*exp(3*x))/3 ).
a(n) = Sum_{k=0..n} (3*k+1)^(n-1) * binomial(n,k) = 2^n * A349714(n).
a(n) ~ sqrt(LambertW(exp(-1)) + 1) * 3^(n-1) * n^(n-1) / (exp(n) * LambertW(exp(-1))^(n + 1/3)). - Vaclav Kotesovec, Apr 24 2024

A362735 E.g.f. satisfies A(x) = exp(x + x / A(x)^2).

Original entry on oeis.org

1, 2, -4, 56, -1008, 25632, -833600, 33067904, -1548418816, 83597525504, -5112566055936, 349330707068928, -26374805535322112, 2180554321981349888, -195926186031705505792, 19010400989418574020608, -1980997069982960384409600, 220651645970702249702326272
Offset: 0

Views

Author

Seiichi Manyama, May 01 2023

Keywords

Links

Crossrefs

Cf. A349562, A362693, A362694, A362734.
Cf. A349720.

Programs

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

Formula

E.g.f.: sqrt( 2*x / LambertW(2*x*exp(-2*x)) ) = exp( x + LambertW(2*x*exp(-2*x))/2 ).
a(n) = Sum_{k=0..n} (-2*k+1)^(n-1) * binomial(n,k) = 2^n * A349720(n).

A362736 E.g.f. satisfies A(x) = exp(x^2 + x / A(x)).

Original entry on oeis.org

1, 1, 1, 4, -3, 96, -755, 10368, -147623, 2492416, -47137959, 996741120, -23260103339, 594198429696, -16492683271259, 494278721929216, -15908038836914895, 547238863907586048, -20038031401448021327, 778147549666716155904, -31943565308583934360019
Offset: 0

Views

Author

Seiichi Manyama, May 01 2023

Keywords

Links

Crossrefs

Cf. A362693, A362737.
Cf. A362690.

Programs

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

Formula

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

A362737 E.g.f. satisfies A(x) = exp(x^3 + x / A(x)).

Original entry on oeis.org

1, 1, -1, 10, -27, 316, -3725, 63666, -1177687, 25196536, -607345209, 16391726110, -488872392371, 15968546353332, -566886190710853, 21733419523383946, -894910999976666415, 39390009619800983536, -1845602126785662907121, 91714859182521808208694
Offset: 0

Views

Author

Seiichi Manyama, May 01 2023

Keywords

Links

Crossrefs

Cf. A362693, A362736.
Cf. A362691.

Programs

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

Formula

E.g.f.: x / LambertW(x*exp(-x^3)) = exp( x^3 + LambertW(x*exp(-x^3)) ).
a(n) = n! * Sum_{k=0..floor(n/3)} (-n+3*k+1)^(n-2*k-1) / (k! * (n-3*k)!).
Showing 1-5 of 5 results.

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

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