A038057 -id:A038057 - cp's OEIS frontend

A052746 a(0) = 0; a(n) = (2*n)^(n-1), n > 0.

0, 1, 4, 36, 512, 10000, 248832, 7529536, 268435456, 11019960576, 512000000000, 26559922791424, 1521681143169024, 95428956661682176, 6502111422497947648, 478296900000000000000, 37778931862957161709568, 3189059870763703892770816, 286511799958070431838109696

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Expansion of inverse of x*exp(2x).
Number of well-colored directed trees on n nodes. Well-colored means, each green vertex has at least a red child, each red vertex has no red child.
Number of labeled rooted directed trees on n nodes.

Robert Israel, Table of n, a(n) for n = 0..350
Federico Ardila, Matthias Beck, and Jodi McWhirter, The Arithmetic of Coxeter Permutahedra, arXiv:2004.02952 [math.CO], 2020.
C. Banderier, J.-M. Le Bars, and V. Ravelomanana, Generating functions for kernels of digraphs, arXiv:math/0411138 [math.CO], 2004.
Theo Douvropoulos, Joel Brewster Lewis, and Alejandro H. Morales, Hurwitz numbers for reflection groups III: Uniform formulas, arXiv:2308.04751 [math.CO], 2023, see p. 11.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 702

Cf. A019762 (2*e), A038057, A097627.

spec := [S,{B=Set(S),S=Prod(Z,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

terms = 19;
A[x_] = -1/2 LambertW[-2 x];
CoefficientList[A[x] + O[x]^terms, x] Range[0, terms-1]! (* Jean-François Alcover, Jan 15 2019 *)
Join[{0},Table[(2n)^(n-1),{n,20}]] (* Harvey P. Dale, Dec 14 2020 *)

a(n)=if(n,(2*n)^(n-1),0) \\ Charles R Greathouse IV, Nov 20 2011

[lucas_number1(n, 2*n, 0) for n in range(0, 17)] # Zerinvary Lajos, Mar 09 2009

E.g.f.: -1/2*W(-2*x), where W is Lambert's W function.
From Robert Israel, Jun 16 2016: (Start)
E.g.f. g(x) satisfies g(x) = x*exp(2*g(x)) and (1-2*g(x)) g'(x) = g(x).
a(n) = (2*n/(n-1)) * Sum_{j=1..n-1} binomial(n-1,j)*a(j)*a(n-j) for n >= 2. (End)
a(n) = [x^n] x/(1 - 2*n*x). - Ilya Gutkovskiy, Oct 12 2017
Limit_{n->oo} a(n+1)/(n*a(n)) = 2*e. - Stefano Spezia, Mar 12 2023

New description from Vladeta Jovovic, Mar 08 2003

A038058 Number of labeled trees with 2-colored nodes.

Original entry on oeis.org

1, 2, 4, 24, 256, 4000, 82944, 2151296, 67108864, 2448880128, 102400000000, 4829076871168, 253613523861504, 14681377947951104, 928873060356849664, 63772920000000000000, 4722366482869645213696, 375183514207494575620096

Offset: 0

Christian G. Bower, Jan 04 1999

nonn

Robert Israel, Table of n, a(n) for n = 0..321
Index entries for sequences related to trees

Cf. A000272, A038055-A038062.

1, seq(2^n * n^(n-2), n=1..20); # Robert Israel, Nov 02 2014

nn = 17; f[x_] := Sum[n^(n - 2) x^n/n!, {n, 1, nn}];
Range[0, nn]! CoefficientList[Series[f[2 x] + 1, {x, 0, nn}], x] (* Geoffrey Critzer, Nov 02 2014 *)

a(n) = A038057(n)/n = 2^n * n^(n-2) for n>=1. E.g.f. B(2*x) where B(x) is e.g.f. of A000272.

A320064 The number of F_2 graphs on { 1, 2, ..., n } with a unique cycle of weight 1, which corresponds to the number of reflectable bases of the root system of type D_n.

Original entry on oeis.org

0, 1, 16, 312, 7552, 220800, 7597824, 301321216, 13545271296, 681015214080, 37879390720000, 2309968030334976, 153275504883695616, 10995166075754119168, 847974316241667686400, 69971459959477921382400, 6151490510604350965940224, 574035430519008722436489216, 56669921387839814123670994944

Offset: 1

Masaya Tomie, Oct 04 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..350
Federico Ardila, Matthias Beck, and Jodi McWhirter, The Arithmetic of Coxeter Permutahedra, arXiv:2004.02952 [math.CO], 2020.
S. Azam, M. B. Soltani, M. Tomie and Y. Yoshii, A graph theoretical classification for reflectable bases, PRIMS, Vol 55 no 4, (2019), 689-736.
Theo Douvropoulos, Joel Brewster Lewis, and Alejandro H. Morales, Hurwitz numbers for reflection groups III: Uniform formulas, arXiv:2308.04751 [math.CO], 2023, see p. 11.

Cf. A000435, A001863, A038057.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&+[(&+[j^(j-1)*(2*x)^j/Factorial(j): j in [1..m+2]])^k/(4*k): k in [2..m+1]]) )); [0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Dec 10 2018

nmax = 20; Rest[CoefficientList[Series[Sum[1/(4*m)*(Sum[k^(k-1)*(2*x)^k/k!, {k, 1, nmax}])^m, {m, 2, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Oct 23 2018 *)

seq(n)={Vec(serlaplace(sum(m=2, n, (sum(k=1, n, k^(k-1)*(2*x)^k/k!) + O(x^n))^m/(4*m))), -n)} \\ Andrew Howroyd, Nov 07 2018

apply( A320064(n)=A001863(n)*(n-1)<<(n-2), [1..20]) \\ Defines the function A320064. The additional apply(...) provides a check and illustration. - M. F. Hasler, Dec 09 2018

from math import comb
def A320064(n): return 0 if n<2 else ((sum(comb(n,k)*(n-k)**(n-k)*k**k for k in range(1,(n+1>>1)))<<1) + (0 if n&1 else comb(n,m:=n>>1)*m**n))//n<Chai Wah Wu, Apr 25-26 2023

E.g.f.: Sum_{m>=2} (1/(4*m)) (Sum_{k>=1} k^(k-1)*(2*x)^k/k!)^m.
a(n) = (n-1)*2^(n-2)*A001863(n). - M. F. Hasler, Dec 09 2018
a(n) = 2^(n-2)*A000435(n). - Chai Wah Wu, Apr 25 2023

A361291 a(n) = ((2n + 1)^n - 1)/(2n).

Original entry on oeis.org

1, 6, 57, 820, 16105, 402234, 12204241, 435984840, 17927094321, 833994048910, 43309534450633, 2483526865641276, 155867505885345241, 10627079738421409410, 782175399728156197665, 61812037545704964935440, 5220088150634922700769761, 469168161404536131943150998

Offset: 1

Stefano Spezia, Mar 12 2023

This sequence is of the form (k^n - 1)/(k - 1) with k = 2*n + 1. See crossrefs in A218722 for other sequences of the same form.

Cf. A005408, A019762 (2*e), A051129, A218722.

Cf. A000169, A000312, A038057, A052746, A062971, A213236.

Mathematica
```
Table[((2n+1)^n-1)/(2n),{n,20}]
```

def A361291(n): return (((n<<1)+1)**n-1)//(n<<1) # Chai Wah Wu, Mar 14 2023

a(n) = Sum_{i=0..n-1} A005408(n)^i.
a(n) = n! * [x^n] exp(x)*(exp(2*n*x) - 1)/(2*n).
a(n) = n! * [x^n] exp((n+1)*x)*sinh(n*x)/n.
Limit_{n->oo} a(n+1)/(n*a(n)) = 2*e.
Limit_{n->oo} (a(n+1)/a(n) - a(n)/a(n-1)) = 2*e.

cp's OEIS Frontend

A052746 a(0) = 0; a(n) = (2*n)^(n-1), n > 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

Extensions

A038058 Number of labeled trees with 2-colored nodes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A320064 The number of F_2 graphs on { 1, 2, ..., n } with a unique cycle of weight 1, which corresponds to the number of reflectable bases of the root system of type D_n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Python

Formula

A361291 a(n) = ((2*n + 1)^n - 1)/(2*n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python

Formula

A361291 a(n) = ((2n + 1)^n - 1)/(2n).