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 12 results. Next

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

A362693 E.g.f. satisfies A(x) = exp(x + x / A(x)).

Original entry on oeis.org

1, 2, 0, 8, -64, 832, -13568, 269824, -6328320, 171044864, -5235245056, 178988498944, -6760886435840, 279614956503040, -12566949343002624, 609881495812702208, -31785828867471572992, 1770660964785178279936, -104990165030126886060032
Offset: 0

Views

Author

Seiichi Manyama, May 01 2023

Keywords

Links

Crossrefs

Cf. A349562, A362694, A362734, A362735.
Cf. A362736, A362737.
Cf. A349719.

Programs

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

Formula

E.g.f.: x / LambertW(x*exp(-x)) = exp( x + LambertW(x*exp(-x)) ).
a(n) = Sum_{k=0..n} (-k+1)^(n-1) * binomial(n,k) = 2^n * A349719(n).

A362472 E.g.f. satisfies A(x) = exp(x + x^3 * A(x)^3).

Original entry on oeis.org

1, 1, 1, 7, 97, 961, 10201, 177241, 3801505, 80718625, 1887205681, 52896262321, 1648697978401, 54216677033377, 1928791931034697, 75326014326206281, 3159713152034201281, 140373558362282197441, 6632746205445950124385, 333591744669464008432225
Offset: 0

Views

Author

Seiichi Manyama, Apr 21 2023

Keywords

Links

Crossrefs

Column k=6 of A362490.
Cf. A143768, A349562, A362473.
Cf. A362392, A362481.

Programs

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

Formula

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

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).

A362473 E.g.f. satisfies A(x) = exp(x + x^4 * A(x)^4).

Original entry on oeis.org

1, 1, 1, 1, 25, 601, 9001, 105001, 1231441, 24146641, 740098801, 22443260401, 607394284201, 16102368745321, 497289446373721, 19072987370400601, 806135144596672801, 33945128330918599201, 1426006261391514829921, 63478993000497055809121
Offset: 0

Views

Author

Seiichi Manyama, Apr 21 2023

Keywords

Links

Crossrefs

Cf. A143768, A349562, A362472.
Cf. A362393, A362482, A362491.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-4*x^4*exp(4*x))/4)))

Formula

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

A362690 E.g.f. satisfies A(x) = exp(x^2 + x * A(x)).

Original entry on oeis.org

1, 1, 5, 28, 245, 2816, 40537, 702976, 14270153, 332102656, 8719631981, 255020847104, 8222803663549, 289815184113664, 11085650268060929, 457386463819595776, 20248713707077863953, 957435459515190345728, 48157934732749633188565
Offset: 0

Views

Author

Seiichi Manyama, May 01 2023

Keywords

Comments

Essentially the same as A138293.

Links

Crossrefs

Cf. A349562, A362691.
Cf. A125500, A138293, A362736.

Programs

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

Formula

E.g.f.: -LambertW(-x * exp(x^2)) / x = 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)!).
a(n) ~ sqrt(1 + LambertW(2*exp(-2))) * 2^((n+1)/2) * n^(n-1) / (exp(n) * LambertW(2*exp(-2))^((n+1)/2)). - Vaclav Kotesovec, Nov 10 2023

A362691 E.g.f. satisfies A(x) = exp(x^3 + x * A(x)).

Original entry on oeis.org

1, 1, 3, 22, 173, 1836, 24847, 403474, 7667865, 167097016, 4108985531, 112562882334, 3399748630357, 112246652293972, 4022094151907847, 155461592488721866, 6447531477912609713, 285606134199075271536, 13458367778796518816755
Offset: 0

Views

Author

Seiichi Manyama, May 01 2023

Keywords

Links

Crossrefs

Cf. A349562, A362690.
Cf. A362737.

Programs

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

Formula

E.g.f.: -LambertW(-x * exp(x^3)) / x = 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)!).

A362747 E.g.f. satisfies A(x) = exp(x^2/2 + x * A(x)).

Original entry on oeis.org

1, 1, 4, 22, 182, 1996, 27412, 453160, 8767516, 194438800, 4864250096, 135538060384, 4163356010728, 139784741268160, 5093269640966704, 200170986137297536, 8440841773833141008, 380153135554220691712, 18212499110682362677312
Offset: 0

Views

Author

Seiichi Manyama, May 02 2023

Keywords

Links

Crossrefs

Cf. A349562, A362748.
Cf. A143740, A362690.

Programs

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

Formula

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

A362748 E.g.f. satisfies A(x) = exp(x^3/6 + x * A(x)).

Original entry on oeis.org

1, 1, 3, 17, 133, 1386, 18097, 284299, 5225985, 110097836, 2616190831, 69236871309, 2019833025157, 64403044165942, 2228441614038837, 83166830262851591, 3330183199746011713, 142418071427679810936, 6478769455582913796475
Offset: 0

Views

Author

Seiichi Manyama, May 02 2023

Keywords

Links

Crossrefs

Cf. A349562, A362747.
Cf. A362381, A362691.

Programs

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

Formula

E.g.f.: -LambertW(-x * exp(x^3/6)) / x = exp( x^3/6 - LambertW(-x*exp(x^3/6)) ).
a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k+1)^(n-2*k-1) / (6^k * k! * (n-3*k)!).
Showing 1-10 of 12 results. Next

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

Content is available under The OEIS End-User License Agreement.