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 1-6 of 6 results.

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).

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).

A372315 Expansion of e.g.f. exp( x - LambertW(-2*x)/2 ).

Original entry on oeis.org

1, 2, 8, 68, 960, 18832, 471136, 14324480, 512733696, 21119803136, 984029612544, 51169331031040, 2937675286583296, 184560174104465408, 12594824112085327872, 927757127285523243008, 73369903633161123397632, 6200198958236463387836416
Offset: 0

Views

Author

Seiichi Manyama, Apr 27 2024

Keywords

Links

Crossrefs

Cf. A088957, A360193, A372316, A372320, A372321.
Cf. A362694.

Programs

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

Formula

a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n,k).
G.f.: Sum_{k>=0} (2*k+1)^(k-1) * x^k / (1-x)^(k+1).
a(n) ~ 2^(n-1) * n^(n-1) * exp((exp(-1) + 1)/2). - Vaclav Kotesovec, May 04 2024

A372235 E.g.f. A(x) satisfies A(x) = exp( x * (1 + A(x)^(3/2)) ).

Original entry on oeis.org

1, 2, 10, 98, 1456, 29132, 734932, 22407464, 801710560, 32940601424, 1528816004944, 79109107128944, 4516145972879680, 281970941337424640, 19114791434098402816, 1398205517746364523008, 109771912847021666795008, 9206931548976575570314496
Offset: 0

Views

Author

Seiichi Manyama, Apr 24 2024

Keywords

Links

Crossrefs

Cf. A349562, A362694, A362734, A372236.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-2/3*lambertw(-3*x/2*exp(3*x/2)))))
  • PARI
    a(n, r=1, t=0, u=3/2) = r*sum(k=0, n, (t*n+u*k+r)^(n-1)*binomial(n, k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (3*k/2+1)^(k-1)*x^k/(1-(3*k/2+1)*x)^(k+1)))

Formula

E.g.f.: A(x) = exp( x - 2/3 * LambertW(-3*x/2 * exp(3*x/2)) ).
If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + A(x)^(u/r)) ), then a(n) = r * Sum_{k=0..n} (t*n+u*k+r)^(n-1) * binomial(n,k).
G.f.: Sum_{k>=0} (3*k/2+1)^(k-1) * x^k/(1 - (3*k/2+1)*x)^(k+1).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (2^(n-1) * exp(n) * LambertW(exp(-1))^(n + 2/3)). - Vaclav Kotesovec, Apr 24 2024

A372278 E.g.f. A(x) satisfies A(x) = exp( x * (1 + A(x)^(5/2)) ).

Original entry on oeis.org

1, 2, 14, 218, 5256, 172332, 7161964, 360849848, 21378442976, 1456505344592, 112197636802224, 9643110922761648, 914870017865191936, 94969006015521439232, 10707303771557931935744, 1302965738334245437242368, 170216425515761065556430336
Offset: 0

Views

Author

Seiichi Manyama, Apr 25 2024

Keywords

Links

Crossrefs

Cf. A349562, A362694, A362734, A372236, A372235.
Cf. A349716, A372279.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-2/5*lambertw(-5*x/2*exp(5*x/2)))))
  • PARI
    a(n, r=1, t=0, u=5/2) = r*sum(k=0, n, (t*n+u*k+r)^(n-1)*binomial(n, k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (5*k/2+1)^(k-1)*x^k/(1-(5*k/2+1)*x)^(k+1)))

Formula

E.g.f.: A(x) = exp( x - 2/5 * LambertW(-5*x/2 * exp(5*x/2)) ).
If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + A(x)^(u/r)) ), then a(n) = r * Sum_{k=0..n} (t*n+u*k+r)^(n-1) * binomial(n,k).
G.f.: Sum_{k>=0} (5*k/2+1)^(k-1) * x^k/(1 - (5*k/2+1)*x)^(k+1).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 5^(n-1) * n^(n-1) / (2^(n-1) * LambertW(exp(-1))^(n + 2/5) * exp(n)). - Vaclav Kotesovec, May 06 2024
Showing 1-6 of 6 results.

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