A009444 -id:A009444 - cp's OEIS frontend

A009306 Expansion of e.g.f.: log(1 + exp(x)*x).

0, 1, 1, -1, -2, 9, 6, -155, 232, 3969, -20870, -118779, 1655028, 1610257, -143697722, 522358005, 13332842416, -138189937791, -1128293525646, 29219838555781, 17274118159180, -5993074252801839, 38541972209299966, 1179892974640047669

Offset: 0

R. H. Hardin

Alois P. Heinz, Table of n, a(n) for n = 0..200
Gottfried Helms, Infinite sum of powerseries likely converges to a powerseries with rational coefficients ... May 14, 2021
Vaclav Kotesovec, Plot of (abs(a(n))/n!)^(1/n) for n = 1..1000
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A009444.

a:= n-> n! *add(k^(n-k-1) *(-1)^(k+1) /(n-k)!, k=1..n):
seq(a(n), n=0..25);

With[{nn=30},CoefficientList[Series[Log[1+Exp[x]x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 22 2016 *)

seq(n)=Vec(serlaplace(log(1 + exp(x + O(x^n))*x)), -(n+1)) \\ Andrew Howroyd, May 26 2021

a(n) = n! * Sum_{k=1..n} k^(n-k-1) * (-1)^(k+1)/(n-k)!. - Vladimir Kruchinin, Sep 07 2010
a(n) = n - Sum_{k=1..n-1} binomial(n-1,k-1) * (n-k) * a(k). - Ilya Gutkovskiy, Jan 17 2020
Lim sup_{n->infinity} (abs(a(n))/n!)^(1/n) = 1/abs(LambertW(-1)) = 1/A238274. - Vaclav Kotesovec, May 26 2021

Extended with signs by Olivier Gérard, Mar 15 1997

Definition clarified by Harvey P. Dale, Oct 22 2016

A305990 Expansion of e.g.f.: (1+x) / (exp(-x) - x).

Original entry on oeis.org

1, 3, 11, 58, 409, 3606, 38149, 470856, 6641793, 105398650, 1858413061, 36044759796, 762659322385, 17481598316742, 431535346662645, 11413394655983536, 321989729198400385, 9651573930139850610, 306321759739045148293, 10262156907184058219340

Offset: 0

Vaclav Kotesovec, Jun 16 2018

nonn

Cf. A305133, A006153, A009306, A009444, A072597, A089148, A302397.

nmax = 20; CoefficientList[Series[(1+x)/(E^(-x)-x), {x, 0, nmax}], x] * Range[0, nmax]!
a={1};For[n=1,n<20,n++,AppendTo[a,Sum[(n!)*((n-k+1)^(k-1))*(n+1)/(k!),{k,0,n+1}]]]; a (* Detlef Meya, Sep 05 2023 *)

a(n) ~ n! / LambertW(1)^(n+1).
a(n) = (-1)^n * A009444(n+1).
a(n) = Sum_{k=0..n+1} (n+1)!*(n-k+1)^(k-1)/k! for n > 0. - Detlef Meya, Sep 05 2023

A346753 Expansion of e.g.f. -log( 1 - x^2 * exp(x) / 2 ).

Original entry on oeis.org

0, 0, 1, 3, 9, 40, 225, 1491, 11578, 102852, 1026945, 11394955, 139091106, 1852061718, 26716291693, 415033647315, 6908006807640, 122645325067576, 2313546734841633, 46209268921868595, 974228913850588750, 21620679147700290210, 503810188866302511501

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

nonn

Cf. A000217, A009444, A133189, A292953, A305990, A346750, A346754, A346755.

nmax = 22; CoefficientList[Series[-Log[1 - x^2 Exp[x]/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = Binomial[n, 2] + (1/n) Sum[Binomial[n, k] Binomial[n - k, 2] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 22}]

a(0) = 0; a(n) = binomial(n,2) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,2) * k * a(k).
a(n) ~ (n-1)! / (2*LambertW(1/sqrt(2)))^n. - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=1..floor(n/2)} k^(n-2*k-1)/(2^k * (n-2*k)!). - Seiichi Manyama, Dec 14 2023

A346754 Expansion of e.g.f. -log( 1 - x^3 * exp(x) / 3! ).

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 30, 175, 1176, 7364, 50520, 425205, 4010380, 39433966, 414654604, 4793188855, 59834495280, 789420239560, 11016095913456, 163423065359529, 2565467553034740, 42320595474149650, 732058678770177220, 13275485607004016011

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

nonn

Cf. A000292, A009444, A145453, A292954, A305990, A346751, A346753, A346755.

nmax = 23; CoefficientList[Series[-Log[1 - x^3 Exp[x]/3!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = Binomial[n, 3] + (1/n) Sum[Binomial[n, k] Binomial[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]

a(0) = 0; a(n) = binomial(n,3) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,3) * k * a(k).
a(n) ~ (n-1)! / (3*LambertW(2^(1/3)/3^(2/3)))^n. - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=1..floor(n/3)} k^(n-3*k-1)/(6^k * (n-3*k)!). - Seiichi Manyama, Dec 14 2023

A346755 Expansion of e.g.f. -log( 1 - x^4 * exp(x) / 4! ).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 15, 35, 105, 756, 6510, 46530, 289245, 1892605, 16187171, 170721915, 1833783770, 18875258780, 196470797580, 2255939795436, 29179692064545, 401813199660285, 5612352516200815, 79620308330422475, 1182881543312932386

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

nonn

Cf. A000332, A009444, A145454, A292955, A305990, A346752, A346753, A346754.

nmax = 24; CoefficientList[Series[-Log[1 - x^4 Exp[x]/4!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = Binomial[n, 4] + (1/n) Sum[Binomial[n, k] Binomial[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 24}]

a(0) = 0; a(n) = binomial(n,4) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,4) * k * a(k).
a(n) ~ (n-1)! / (4*LambertW(3^(1/4)/2^(5/4)))^n. - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=1..floor(n/4)} k^(n-4*k-1)/(24^k * (n-4*k)!). - Seiichi Manyama, Dec 14 2023

A366459 Expansion of e.g.f. -log(1 - x^2 * exp(x)).

Original entry on oeis.org

0, 0, 2, 6, 24, 140, 990, 8442, 84056, 955656, 12227130, 173812430, 2717859012, 46362339036, 856770362630, 17050946225250, 363576478312560, 8269357341437072, 199837364514425586, 5113346326011170838, 138106722548779770620, 3926456810081828991780

Offset: 0

Seiichi Manyama, Dec 14 2023

nonn

Cf. A009444, A366546.

Cf. A216507, A346753, A358080.

a(n) = n!*sum(k=1, n\2, k^(n-2*k-1)/(n-2*k)!);

a(n) = n! * Sum_{k=1..floor(n/2)} k^(n-2*k-1)/(n-2*k)!.

a(n) ~ (n-1)! / (2^n *LambertW(1/2)^n). - Vaclav Kotesovec, Dec 29 2023

A228534 Triangular array read by rows: T(n,k) is the number of functional digraphs on {1,2,...,n} such that every element is mapped to a recurrent element and there are exactly k cycles, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 3, 1, 11, 9, 1, 58, 71, 18, 1, 409, 620, 245, 30, 1, 3606, 6274, 3255, 625, 45, 1, 38149, 73339, 45724, 11795, 1330, 63, 1, 470856, 977780, 697004, 221529, 33880, 2506, 84, 1, 6641793, 14678712, 11602394, 4309956, 823179, 82908, 4326, 108, 1

Offset: 1

Geoffrey Critzer, Aug 24 2013

The Bell transform of (-1)^n*A009444(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			       1;
       3,      1;
      11,      9,      1;
      58,     71,     18,      1;
     409,    620,    245,     30,     1;
    3606,   6274,   3255,    625,    45,    1;
   38149,  73339,  45724,  11795,  1330,   63,  1;
  470856, 977780, 697004, 221529, 33880, 2506, 84, 1;

Alois P. Heinz, Rows n = 1..90, flattened

Row sums = A006153.
Column 1 = |A009444|.
Cf. A199673.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
g := n -> add(m^(n-m)*m!*binomial(n+1,m), m=1..n+1);
BellMatrix(g, 9); # Peter Luschny, Jan 29 2016

nn = 8; a = x Exp[x];
Map[Select[#, # > 0 &] &,
  Drop[Range[0, nn]! CoefficientList[
     Series[1/(1 - a)^y, {x, 0, nn}], {x, y}], 1]] // Grid
(* Second program: *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, (n+1)! Sum[m^(n-m)/(n-m+1)!, {m, 1, n+1}]], rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

# uses[bell_matrix from A264428, A009444]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: (-1)^n*A009444(n+1), 10) # Peter Luschny, Jan 18 2016

E.g.f.: 1/(1 - x*exp(x))^y.

A275385 Number of labeled functional digraphs on n nodes with only odd sized cycles and such that every vertex is at a distance of at most 1 from a cycle.

Original entry on oeis.org

1, 1, 3, 12, 73, 580, 5601, 63994, 844929, 12647016, 211616065, 3914510446, 79320037281, 1747219469164, 41569414869633, 1062343684252530, 29023112392093441, 844101839207139280, 26038508978625589377, 849150487829425227094, 29189561873274715264545

Offset: 0

Geoffrey Critzer, Jul 25 2016

nonn

Equivalently, these are the functions counted by A116956 with the additional constraint that every element is mapped to a recurrent element. A recurrent element is an element on a cycle in the functional digraph.

Alois P. Heinz, Table of n, a(n) for n = 0..411

Cf. A000246, A006153, A009444, A116956, A216401.

b:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,
       (j-1)!*b(n-j)*binomial(n-1, j-1), 0), j=1..n))
    end:
a:= n-> add(b(j)*j^(n-j)*binomial(n, j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 25 2016

nn = 20; Range[0, nn]! CoefficientList[Series[Sqrt[(1 + z*Exp[z])/(1 - z*Exp[z])], {z, 0, nn}], z]

default(seriesprecision, 30);
S=sqrt((1 + x*exp(x))/(1 - x*exp(x)));
v=Vec(S); for(n=2,#v-1,v[n+1]*=n!); v \\ Charles R Greathouse IV, Jul 29 2016

E.g.f.: sqrt((1 + z*exp(z))/(1 - z*exp(z))).
Exponential transform of A216401.
a(n) ~ 2 * n^n / (sqrt(1+LambertW(1)) * LambertW(1)^n * exp(n)). - Vaclav Kotesovec, Jun 26 2022

A307125 Expansion of e.g.f. log(1 - log(1 - x*exp(x))).

Original entry on oeis.org

0, 1, 2, 4, 17, 123, 1052, 10568, 125750, 1726189, 26730394, 460982300, 8766443952, 182229703043, 4110207945794, 99970680376908, 2608221938476016, 72656914458625593, 2152355976206481570, 67562405794276542004, 2240111797037473955984, 78229640115171735522015, 2870092624821982184377202

Offset: 0

Ilya Gutkovskiy, Mar 26 2019

nonn

Cf. A007550, A009444, A089064, A307126.

a:=series(log(1-log(1-x*exp(x))),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Apr 03 2019

nmax = 22; CoefficientList[Series[Log[1 - Log[1 - x Exp[x]]], {x, 0, nmax}], x] Range[0, nmax]!

my(x = 'x + O('x^30)); concat(0, Vec(serlaplace(log(1 - log(1 - x*exp(x)))))) \\ Michel Marcus, Mar 26 2019

A366546 Expansion of e.g.f. -log(1 - x^3 * exp(x)).

Original entry on oeis.org

0, 0, 0, 6, 24, 60, 480, 5250, 40656, 363384, 4839120, 65198430, 859543080, 13311494196, 233478687624, 4190929145130, 79746180437280, 1667320408619760, 36965002127643936, 854734007793179574, 20962277675893792440, 544839141515795731500

Offset: 0

Seiichi Manyama, Dec 14 2023

nonn

Cf. A009444, A366459.

Cf. A292889, A346754, A358081.

a(n) = n!*sum(k=1, n\3, k^(n-3*k-1)/(n-3*k)!);

a(n) = n! * Sum_{k=1..floor(n/3)} k^(n-3*k-1)/(n-3*k)!.

cp's OEIS Frontend

A009306 Expansion of e.g.f.: log(1 + exp(x)*x).

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A305990 Expansion of e.g.f.: (1+x) / (exp(-x) - x).

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A346753 Expansion of e.g.f. -log( 1 - x^2 * exp(x) / 2 ).

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A346754 Expansion of e.g.f. -log( 1 - x^3 * exp(x) / 3! ).

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A346755 Expansion of e.g.f. -log( 1 - x^4 * exp(x) / 4! ).

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A366459 Expansion of e.g.f. -log(1 - x^2 * exp(x)).

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A228534 Triangular array read by rows: T(n,k) is the number of functional digraphs on {1,2,...,n} such that every element is mapped to a recurrent element and there are exactly k cycles, n>=1, 1<=k<=n.

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A275385 Number of labeled functional digraphs on n nodes with only odd sized cycles and such that every vertex is at a distance of at most 1 from a cycle.

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A307125 Expansion of e.g.f. log(1 - log(1 - x*exp(x))).

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