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 1-3 of 3 results.

A362492 E.g.f. satisfies A(x) = exp(x - x^2/2 * A(x)^2).

Original entry on oeis.org

1, 1, 0, -8, -38, 106, 3676, 24508, -296036, -9149156, -56500064, 2211573376, 64958496472, 184823374360, -35372361487280, -971135892546224, 4364710018963216, 1034808592156017424, 25290798052846014208, -474242641154857953152, -49625273567646267051104
Offset: 0

Views

Author

Seiichi Manyama, Apr 22 2023

Keywords

Links

Crossrefs

Cf. A362493, A362494.
Cf. A362480.

Programs

  • Maple
    N:= 50: # for a(0)..a(N)
egf:=  exp(x - LambertW(x^2 * exp(2*x))/2):
S:=series(egf,x,N+1):
[seq](coeff(S,x,i)*i!,i=0..N); # Robert Israel, May 22 2023
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(x^2*exp(2*x))/2)))

Formula

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

A362481 E.g.f. satisfies A(x) = exp(x - x^3 * A(x)^3).

Original entry on oeis.org

1, 1, 1, -5, -95, -959, -5159, 69721, 3113377, 64493857, 654012721, -13761498959, -1013114081759, -32273321679455, -492845589685175, 13357113599586121, 1410278045569310401, 61239473424756703681, 1270682827211021594977, -40402942687262364034463
Offset: 0

Views

Author

Seiichi Manyama, Apr 21 2023

Keywords

Links

Crossrefs

Cf. A362480, A362482.
Cf. A362430, A362472.

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)} (-1)^k * (3*k+1)^(n-2*k-1) / (k! * (n-3*k)!).

A362482 E.g.f. satisfies A(x) = exp(x - x^4 * A(x)^4).

Original entry on oeis.org

1, 1, 1, 1, -23, -599, -8999, -104999, -868559, 5246641, 582598801, 21205760401, 571129277401, 11475082596121, 81837031796521, -7904119577117399, -596529385424263199, -28051840646006771999, -991870986521074646879, -21837506791918601443679
Offset: 0

Views

Author

Seiichi Manyama, Apr 21 2023

Keywords

Links

Crossrefs

Cf. A362480, A362481.
Cf. A362431, A362473.

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)} (-1)^k * (4*k+1)^(n-3*k-1) / (k! * (n-4*k)!).
Showing 1-3 of 3 results.

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

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