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.

A038049 Number of labeled rooted trees with 2-colored leaves.

Original entry on oeis.org

2, 4, 24, 224, 2880, 47232, 942592, 22171648, 600698880, 18422374400, 630897721344, 23864653578240, 988197253808128, 44460603225407488, 2159714024218951680, 112652924603290615808, 6280048587936003784704, 372616014329572403183616, 23445082059018189741752320
Offset: 1

Views

Author

Christian G. Bower, Jan 04 1999

Keywords

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 185 (3.1.83)

Links

Crossrefs

Cf. A000169, A029856, A038050, A038054, A088789.

Programs

  • Maple
    a:= n-> add(binomial(n, k)*(n-k)^(n-1), k=0..n):
seq(a(n), n=1..20);  # Alois P. Heinz, Nov 30 2012
  • Mathematica
    Table[n!*Sum[2^j/j!*StirlingS2[n-1,n-j],{j,1,n}],{n,1,20}] (* Vaclav Kotesovec, Nov 30 2012 *)

Formula

Divides by n and shifts left under exponential transform.
E.g.f.: A(x) = x-LambertW(-x*exp(x)). - Vladeta Jovovic, Mar 08 2003
a(n) = Sum_{k=0..n} (binomial(n, k)*(n-k)^(n-1)).
A(x) = 2*compositional inverse of 2*x/(1+exp(2*x)). - Peter Bala, Oct 14 2011
a(n) ~ n^(n-1) * sqrt((1+LambertW(1/e))) / (e*LambertW(1/e))^n. - Vaclav Kotesovec, Nov 30 2012

A029857 Number of rooted trees with 3-colored leaves.

Original entry on oeis.org

3, 3, 9, 28, 94, 328, 1197, 4486, 17235, 67429, 267932, 1078003, 4383784, 17987897, 74385984, 309694232, 1297037177, 5460726214, 23098296648, 98113995068, 418335662448, 1789814398035, 7681522429474, 33061825858259, 142674028869587, 617180102839217
Offset: 1

Views

Author

Christian G. Bower

Keywords

Links

Crossrefs

Cf. A000081, A029856, A038050.

Programs

  • Maple
    with(numtheory): a:= proc(n) option remember; local d,j; if n<=1 then 3*n else (add(d*a(d), d=divisors(n-1)) +add(add(d*a(d), d=divisors(j)) *a(n-j), j=1..n-2))/ (n-1) fi end: seq(a(n), n=1..30); # Alois P. Heinz, Sep 06 2008
  • Mathematica
    a[n_] := a[n] = If[n<=1, 3*n, (Sum[d*a[d], {d, Divisors[n-1]}] + Sum[Sum[ d*a[d], {d, Divisors[j]}]*a[n-j], {j, 1, n-2}])/(n-1)]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 21 2016 *)

Formula

Shifts left under Euler transform.
a(n) ~ c * d^n / n^(3/2), where d = 4.58859196701042554480382685... and c = 0.5102557157321640697473838... - Vaclav Kotesovec, Mar 29 2014
G.f. A(x) satisfies: A(x) = 2*x + x * exp( Sum_{k>=1} A(x^k) / k ). - Ilya Gutkovskiy, May 19 2023

A360465 E.g.f. satisfies A(x) = exp(x * exp(2*x) * A(x)).

Original entry on oeis.org

1, 1, 7, 64, 829, 14056, 295399, 7426252, 217637305, 7291538704, 275050426411, 11540336658676, 533224609095061, 26908386824872216, 1472691380336896399, 86892807951798473116, 5498668489586321670769, 371511527654280649783840
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2023

Keywords

Links

Crossrefs

Cf. A038050, A273954, A357247, A360466.

Programs

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

Formula

E.g.f.: A(x) = exp( -LambertW(-x * exp(2*x)) ).
E.g.f.: A(x) = -LambertW(-x * exp(2*x)) / (x * exp(2*x)).
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x * exp(2*x))^k / k!.
a(n) = Sum_{k=0..n} (2*k)^(n-k) * (k+1)^(k-1) * binomial(n,k).
a(n) ~ sqrt(1+LambertW(2*exp(-1))) * 2^n * n^(n-1) / (exp(n-1) * LambertW(2*exp(-1))^n). - Vaclav Kotesovec, Feb 08 2023
Showing 1-3 of 3 results.

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

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