A029768 -id:A029768 - cp's OEIS frontend

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'.

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

A264436 Triangle read by rows, inverse Bell transform of the complementary Bell numbers (A000587); T(n,k) for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 3, 1, 0, 14, 15, 6, 1, 0, 89, 100, 45, 10, 1, 0, 716, 834, 405, 105, 15, 1, 0, 6967, 8351, 4284, 1225, 210, 21, 1, 0, 79524, 97596, 52220, 16009, 3080, 378, 28, 1, 0, 1041541, 1303956, 721674, 233268, 48699, 6804, 630, 36, 1

Offset: 0

Peter Luschny, Dec 01 2015

			Triangle starts:
1,
0,     1,
0,     1,     1,
0,     3,     3,     1,
0,    14,    15,     6,     1,
0,    89,   100,    45,    10,    1,
0,   716,   834,   405,   105,   15,   1,
0,  6967,  8351,  4284,  1225,  210,  21,  1,
0, 79524, 97596, 52220, 16009, 3080, 378, 28, 1

Peter Luschny, The Bell transform

Cf. A000587, A007549, A029768, A264428, A264429.

# uses[bell_transform from A264428, inverse_bell_transform from A264429]
def A264436_matrix(dim):
    uno = [1]*dim
    complementary_bell_numbers = [sum((-1)^n*b for (n, b) in enumerate (bell_transform(n, uno))) for n in (0..dim)]
    return inverse_bell_transform(dim, complementary_bell_numbers)
A264436_matrix(9)

Row sums are A029768(n-1) for n>=1.

T(n,1) = A007549(n) for n>=1.

A272055 Decimal expansion of -1/(e^2 Ei(-1)), an increasing rooted tree enumeration constant associated with the Euler-Gompertz constant, where Ei is the exponential integral.

Original entry on oeis.org

6, 1, 6, 8, 8, 7, 8, 4, 8, 2, 8, 0, 7, 2, 7, 0, 7, 1, 4, 4, 4, 9, 3, 8, 3, 4, 5, 6, 6, 2, 2, 8, 5, 4, 9, 3, 5, 2, 4, 9, 0, 0, 5, 6, 9, 3, 3, 1, 6, 8, 8, 1, 7, 8, 6, 5, 6, 6, 1, 0, 3, 3, 2, 3, 1, 9, 1, 4, 3, 7, 2, 4, 2, 5, 1, 5, 4, 7, 6, 7, 2, 7, 3, 0, 3, 3, 9, 8, 2, 5, 6, 0, 3, 1, 4, 9, 4, 8, 3, 4, 5, 1, 1

Offset: 0

Jean-François Alcover, Apr 19 2016

			0.61688784828072707144493834566228549352490056933168817865661...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6 Otter's tree enumeration constants, p. 303.

G. C. Greubel, Table of n, a(n) for n = 0..10000
F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer-Verlag, 1992, pp. 24-48.
Eric Weisstein's MathWorld, Gompertz Constant
Eric Weisstein's MathWorld, Exponential Integral
Eric Weisstein's MathWorld, Rooted Tree
Index entries for sequences related to mobiles

Cf. A029768, A073003.

RealDigits[-1/(E^2*ExpIntegralEi[-1]), 10, 103][[1]]

default(realprecision, 100); 1/(exp(2)*eint1(1)) \\ G. C. Greubel, Sep 07 2018

Equals 1 / (e * A073003).

Also equals -1 / (e^2 * (gamma - Sum_{n>=1} (-1)^(n-1)/(n*n!))), where gamma is the Euler-Mascheroni constant A001620.

cp's OEIS Frontend

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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A264436 Triangle read by rows, inverse Bell transform of the complementary Bell numbers (A000587); T(n,k) for n>=0 and 0<=k<=n.

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A272055 Decimal expansion of -1/(e^2 Ei(-1)), an increasing rooted tree enumeration constant associated with the Euler-Gompertz constant, where Ei is the exponential integral.

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