A007889 -id:A007889 - cp's OEIS frontend

A013499 a(n) = 2*n^n, n >= 2, otherwise a(n) = 1.

1, 1, 8, 54, 512, 6250, 93312, 1647086, 33554432, 774840978, 20000000000, 570623341222, 17832200896512, 605750213184506, 22224013651116032, 875787780761718750, 36893488147419103232, 1654480523772673528354

Offset: 0

N. J. A. Sloane

nonn

For n>=1, a(n) gives the number of alternating plane trees (trees such that the son of each vertex is ordered) on the set of vertices {1,2,..,n+1} (Chauve et al.). See A007889 for the non-ordered case. - Peter Bala, Aug 30 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300
C. Chauve, S. Dulucq and A. Rechnitzer, Enumerating alternating trees, J. Combin. Theory Ser. A 94 (2001), 142-151.
Frank Schmidt and Rodica Simion, Card shuffling and a transformation on S_n, Aequationes Math. 44 (1992), no. 1, 11-34.

Cf. A007889.

lst={};Do[a=n^n;b=(n+1)^(n+1);ab=b-a;ba=b+a;AppendTo[lst,ba-ab],{n,4!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 20 2009 *)

a(n) = if (n<=1, 1, 2*n^n); \\ Michel Marcus, Jul 26 2017

For n>1, resultant of x^n+1 and n(x-1). - Ralf Stephan, Nov 20 2004

Name edited by Michel Marcus, Jul 26 2017

A372236 E.g.f. A(x) satisfies A(x) = exp( x * (1 + A(x)^(1/2)) ).

Original entry on oeis.org

1, 2, 6, 26, 152, 1132, 10300, 111064, 1387104, 19713104, 314350064, 5560881328, 108110428288, 2291750937088, 52618613073408, 1301031907140608, 34470409922547200, 974354631630161152, 29270099764874881792, 931275451933870415104, 31285710787985504633856

Offset: 0

Seiichi Manyama, Apr 23 2024

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A007889, A371524.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-2*lambertw(-x/2*exp(x/2)))))

a(n, r=1, t=0, u=1/2) = r*sum(k=0, n, (t*n+u*k+r)^(n-1)*binomial(n, k));

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (k/2+1)^(k-1)*x^k/(1-(k/2+1)*x)^(k+1)))

E.g.f.: A(x) = exp( x - 2*LambertW(-x/2 * exp(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} (k/2+1)^(k-1) * x^k/(1 - (k/2+1)*x)^(k+1).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (2^(n-1) * exp(n) * LambertW(exp(-1))^(n+2)). - Vaclav Kotesovec, Apr 24 2024

A029847 Gessel-Stanley triangle read by rows: triangle of coefficients of polynomials arising in connection with enumeration of intransitive trees by number of nodes and number of right nodes.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 17, 17, 1, 1, 49, 146, 49, 1, 1, 129, 922, 922, 129, 1, 1, 321, 4887, 11234, 4887, 321, 1, 1, 769, 23151, 106439, 106439, 23151, 769, 1, 1, 1793, 101488, 856031, 1679494, 856031, 101488, 1793, 1, 1, 4097, 420512, 6137832, 21442606, 21442606, 6137832, 420512, 4097, 1

Offset: 0

N. J. A. Sloane

For precise definition see Knuth (1997).

Named after the American mathematicians Ira Martin Gessel (b. 1951) and Richard Peter Stanley (b. 1944). - Amiram Eldar, Jun 11 2021

			Triangle begins:
  1;
  .   1;
  .   1,   1;
  .   1,   5,   1;
  .   1,  17,  17,   1;
  .   1,  49, 146,  49,   1;
  .   1, 129, 922, 922, 129, 1;
  .   ...

Alois P. Heinz, Rows n = 0..141, flattened
Donald E. Knuth, Letter to Daniel Ullman and others, Apr 29 1997. [Annotated scanned copy, with permission]
Alexander Postnikov, Intransitive Trees, J. Combin. Theory Ser. A, Vol. 79, No. 2 (1997), pp. 360-366.

Row sums give A007889.

f:= proc(n,k) option remember; `if`(k<0, 0, `if`(n=0
      and k=0, 1, f(n-1,k-1)+add(add(binomial(n-1, l)
      *s*f(l,s)*f(n-l-1,k-s), s=1..l), l=1..n-1)))
    end:
seq(seq(f(n, k), k=min(n, 1)..n), n=0..10); # Alois P. Heinz, Sep 24 2019

f[n_, k_] := f[n, k] = If[k<0, 0, If[n==0 && k==0, 1, f[n-1, k-1]+Sum[Sum[ Binomial[n-1, l]*s*f[l, s]*f[n-l-1, k-s], {s, 1, l}], {l, 1, n-1}]]];
Table[Table[f[n, k], {k, Min[n, 1], n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 14 2021, after Alois P. Heinz *)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 23 2003

A283828 Number of bounded regions in the Linial arrangement L_{n-1}.

Original entry on oeis.org

0, 0, 1, 4, 26, 212, 2108, 24720, 334072, 5112544, 87396728, 1650607040, 34132685120, 767025716736, 18612106195456, 485013257865472, 13509071081429888, 400505695457942528, 12592502771190979712, 418524228123134068224, 14661145374751901317888

Offset: 1

N. J. A. Sloane, Mar 19 2017

nonn

Except for the initial 0, these are the absolute values of A349719. - Ira M. Gessel, Nov 01 2023

Richard P. Stanley, Hyperplane arrangements, interval orders, and trees, Proc. Natl. Acad. Sci. USA 93 (1996), 2620-2625.
Vasu Tewari, Gessel polynomials, rooks, and extended Linial arrangements, arXiv preprint arXiv:1604.06894 [math.CO], 2016.

Cf. A007889, A349719.

From Ira M. Gessel, Nov 01 2023: (Start)
a(n) = (1/2^n) * Sum_{k=0..n} (k-1)^(n-1) * binomial(n,k) for n>=2.
In the following generating functions we take a(1)=1 rather than a(1)=0.
E.g.f.: 1 + (1/2)*x/LambertW(-(1/2)*x*exp(x/2)).
E.g.f.: 1-1/B(x), where B(x) is the e.g.f. of A007889. See  Corollary 4.2 of Stanley's paper. (End)
a(n) ~ sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (exp(n) * 2^n * LambertW(exp(-1))^(n-1)). - Vaclav Kotesovec, Nov 13 2023

More terms from Ira M. Gessel, Nov 01 2023

A125033 Number of labeled nodes in generation n of a rooted tree where a node with label k has k child nodes with distinct labels, such that each child node is assigned the least unused label in the path that leads back to the root node with label '1'.

Original entry on oeis.org

1, 1, 2, 7, 36, 248, 2157, 22761, 283220, 4068411, 66367834, 1213504295, 24606397802

Offset: 0

Paul D. Hanna and R. J. Mathar, Nov 17 2006

nonn

The minimum label in generation n is n+1 and the maximum label is n*(n+1)/2 + 1.

Compare to trees A029768, A007889, which seem somewhat similar; A029768 has the additional constraint that the labels must be increasing.

			Labels for the initial nodes of the tree for generations 0..4:
gen 0: [1];
gen 1: (1)->[2];
gen 2: (2)->[3,4];
gen 3: (3)->[4,5,6], (4)->[3,5,6,7];
gen 4: (4)->[5,6,7,8], (5)->[4,6,7,8,9], (6)->[4,5,7,8,9,10],
(3)->[5,6,7], (5)->[3,6,7,8,9], (6)->[3,5,7,8,9,10],
(7)->[3,5,6,8,9,10,11];

Alec Mihailovs, A125033: Maple Program (non-recursive, uses little memory); Mapleprimes, Nov 19 2006.

gen := proc(parents,maxgen,ocounts,lvl)
local thislbl,lbl,childlbl,counts,npar;
counts := ocounts;
counts[lvl] := counts[lvl]+1;
if nops(parents) < maxgen then
thislbl := op(-1,parents);
childlbl := 1;
for lbl from 1 to thislbl do
while ( childlbl in parents ) or ( childlbl = thislbl ) do
childlbl := childlbl+1;
od;
npar := [op(parents),childlbl];
if nops(counts) < lvl+1 then
counts := [op(counts),0];
fi;
counts := gen(npar,maxgen,counts,lvl+1);
childlbl := childlbl+1;
od;
fi;
if lvl <= maxgen -4 then
print(counts);
fi;
RETURN(counts);
end:
maxgen := 8;
parents := [1,2];
n := [1,0];
gen(parents,maxgen,n,2);
print(%) ; # R. J. Mathar, Nov 17 2006)
The following Maple10 code is from Alec Mihailovs: # (start)
f:=proc(n::integer[4],A::Array(datatype=integer[4]),
B::Array(datatype=integer[4]))::integer[4]; local c::integer[4],
i::integer[4],len::integer[4],m::integer[4]; c,len,m:=0,3,3;
while len>1 do if len=n then c:=c+1;m:=A[len];B[m]:=0;len:=len-1;
B[A[len]]:=B[A[len]]+1 elif B[A[len]]<=A[len] then for i from m+1
do if B[i]=0 then break fi od; len:=len+1;A[len]:=i;B[i]:=1;m:=2
else m:=A[len];B[m]:=0;len:=len-1;B[A[len]]:=B[A[len]]+1 fi od; c end:
cf:=Compiler:-Compile(f):
F:=proc(n::posint) local A,B;
if n<3 then 1 elif n=3 then 2 else
A:=Array([$1..3,0$(n-3)],datatype=integer[4]);
B:=Array([1$3,0$((n-2)*(n+1)/2)],datatype=integer[4]);
cf(n,A,B) fi end:
seq(F(n),n=1..12); # (end)

Terms a(6)-a(10) from R. J. Mathar, Nov 17 2006

a(11) and a(12) from Alec Mihailovs (alec(AT)mihailovs.com), Nov 19 2006

A140983 E.g.f. is reversion of (2(1+x)log(1+x)+x^2+2x)/( (2+x)^2(1+x) ).

Original entry on oeis.org

1, 3, 17, 145, 1663, 24031, 419521, 8592417, 202069759, 5367258479, 158934860321, 5191969220945, 185490468312767, 7194912503747775, 301130097048242561, 13526711564792340289, 649121580063333263359, 33142745983169890692559

Offset: 1

Brian Drake, Jul 28 2008

a(n) is the number of labeled incomplete ternary trees on n vertices in which each left or middle child has a larger label than its parent and each right child has a smaller label than its parent. For example, a(2)=3 because we have 2L1, 2M1 and 1R2. Here aLb means a is a left child of b, etc.

Alois P. Heinz, Table of n, a(n) for n = 1..100

Cf. A007889.

N:= 8: exp(RootOf(2*_Z*exp(_Z)-x*exp(_Z)-2*x*exp(_Z)^2-x*exp(_Z)^3 -1 +exp(_Z)^2))-1: series(%, x, N+1): convert(%, polynom): seq( i!*coeff(%, x, i), i=1..N);

A371524 E.g.f. A(x) satisfies A(x) = exp( 2 * x * (1 + A(x)^(1/4)) ).

Original entry on oeis.org

1, 4, 20, 124, 936, 8424, 88648, 1072432, 14702720, 225692128, 3839770656, 71780577312, 1463532416320, 32337850727680, 770039603953664, 19664621381714944, 536234348295180288, 15554459021934423552, 478297493455731968512, 15543431292269887979008

Offset: 0

Seiichi Manyama, Apr 23 2024

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A007889, A372236.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(2*x-4*lambertw(-x/2*exp(x/2)))))

a(n, r=2, t=0, u=1/2) = r*sum(k=0, n, (t*n+u*k+r)^(n-1)*binomial(n, k));

my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (k/2+2)^(k-1)*x^k/(1-(k/2+2)*x)^(k+1)))

E.g.f.: A(x) = exp( 2*x - 4*LambertW(-x/2 * exp(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.: 2 * Sum_{k>=0} (k/2+2)^(k-1) * x^k/(1 - (k/2+2)*x)^(k+1).
a(n) ~ sqrt(LambertW(exp(-1)) + 1) * n^(n-1) / (2^(n-2) * exp(n) * LambertW(exp(-1))^(n+4)). - Vaclav Kotesovec, Apr 24 2024

cp's OEIS Frontend

A013499 a(n) = 2*n^n, n >= 2, otherwise a(n) = 1.

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A372236 E.g.f. A(x) satisfies A(x) = exp( x * (1 + A(x)^(1/2)) ).

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A029847 Gessel-Stanley triangle read by rows: triangle of coefficients of polynomials arising in connection with enumeration of intransitive trees by number of nodes and number of right nodes.

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Maple

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A283828 Number of bounded regions in the Linial arrangement L_{n-1}.

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A125033 Number of labeled nodes in generation n of a rooted tree where a node with label k has k child nodes with distinct labels, such that each child node is assigned the least unused label in the path that leads back to the root node with label '1'.

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A140983 E.g.f. is reversion of (2(1+x)log(1+x)+x^2+2x)/( (2+x)^2(1+x) ).

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Programs

Maple

A371524 E.g.f. A(x) satisfies A(x) = exp( 2 * x * (1 + A(x)^(1/4)) ).

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PARI

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