A002104 -id:A002104 - cp's OEIS frontend

A353548 Expansion of e.g.f. -log(1-4x) exp(x)/4.

0, 1, 6, 47, 540, 8429, 166210, 3952955, 109981816, 3502905369, 125648153278, 5011458069639, 219987094389524, 10538817637744005, 547118005892177018, 30595552548140425747, 1833501625083035349488, 117219490267316310468913

Offset: 0

Seiichi Manyama, May 27 2022

nonn

Cf. A002104, A353546, A353547, A353549.
Cf. A346396.
Essentially partial sums of A056545.

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-log(1-4*x)*exp(x)/4)))

a(n) = n!*sum(k=0, n-1, 4^(n-1-k)/((n-k)*k!));

a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(4*i-3)*v[i]-4*(i-1)*v[i-1]+1); v;

a(n) = n! * Sum_{k=0..n-1} 4^(n-1-k) / ((n-k) * k!).
a(0) = 0, a(1) = 1, a(n) = (4 * n - 3) * a(n-1) - 4 * (n-1) * a(n-2) + 1.
a(n) ~ (n-1)! * exp(1/4) * 4^(n-1). - Vaclav Kotesovec, Jun 08 2022

A353549 Expansion of e.g.f. log(1+3x) exp(x)/3.

Original entry on oeis.org

0, 1, -1, 12, -104, 1289, -19605, 356488, -7541464, 182009385, -4935863537, 148600324124, -4918093868688, 177482897072545, -6936155749635541, 291836667412104072, -13152940374866178512, 632196357654491385521, -32280617841842744380161

Offset: 0

Seiichi Manyama, May 27 2022

sign

Cf. A002104, A353546, A353547, A353548, A354419.

Cf. A346398.

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(log(1+3*x)*exp(x)/3)))

a(n) = n!*sum(k=0, n-1, (-3)^(n-1-k)/((n-k)*k!));

a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(-3*i+4)*v[i]+3*(i-1)*v[i-1]+1); v;

a(n) = n! * Sum_{k=0..n-1} (-3)^(n-1-k) / ((n-k) * k!).
a(0) = 0, a(1) = 1, a(n) = (-3 * n + 4) * a(n-1) + 3 * (n-1) * a(n-2) + 1.
a(n) ~ -(-1)^n * (n-1)! * 3^(n-1) / exp(1/3). - Vaclav Kotesovec, Jun 08 2022

A121726 Sum sequence A000522 then subtract 0,1,2,3,4,5,...

Original entry on oeis.org

1, 2, 6, 21, 85, 410, 2366, 16065, 125665, 1112074, 10976174, 119481285, 1421542629, 18348340114, 255323504918, 3809950976993, 60683990530209, 1027542662934898, 18430998766219318, 349096664728623317, 6962409983976703317, 145841989688186383338, 3201192743180799343822

Offset: 1

Alford Arnold, Aug 17 2006

Let aut(p) denote the size of the centralizer of the partition p (see A339016 for the definition). Then a(n) = Sum_{p in P} n!/aut(p), where P are the partitions of n with largest part k and length n + 1 - k. - Peter Luschny, Nov 19 2020

			A000522 begins     1 2 5 16 65 326 ...
with sums          1 3 8 24 89 415 ...
so sequence begins 1 2 6 21 85 410 ...
.
From _Peter Luschny_, Nov 19 2020: (Start):
The combinatorial interpretation is illustrated by this computation of a(5):
5! / aut([5])             = 120 / A339033(5, 1) = 120/5   = 24
5! / aut([4, 1])          = 120 / A339033(5, 2) = 120/4   = 30
5! / aut([3, 1, 1])       = 120 / A339033(5, 3) = 120/6   = 20
5! / aut([2, 1, 1, 1])    = 120 / A339033(5, 4) = 120/12  = 10
5! / aut([1, 1, 1, 1, 1]) = 120 / A339033(5, 5) = 120/120 =  1
--------------------------------------------------------------
                                                Sum: a(5) = 85
(End)

Also the row sums of A092271.

Cf. A000522, A006231, A002104, A339016, A339033.

f[list_] :=Total[list]!/Apply[Times, list]/Apply[Times, Map[Length, Split[list]]!]; Table[Total[Map[f, Select[Partitions[n], Count[#, Except[1]] == 1 &]]] + 1, {n, 1, 20}] (* Geoffrey Critzer, Nov 07 2015 *)

A000522(n)={ return( sum(k=0,n,n!/k!)) ; } A121726(n)={ return(sum(k=0,n-1,A000522(k))-n+1) ; } { for(n=1,25, print1(A121726(n),",") ; ) ; } \\ R. J. Mathar, Sep 02 2006

def A121726(n):
    def h(n, k):
        if n == k: return 1
        return factorial(n)//((n + 1 - k)*factorial(k - 1))
    return sum(h(n, k) for k in (1..n))
print([A121726(n) for n in (1..23)])
# Demonstrates the combinatorial view:
def A121726(n):
    if n == 0: return 1
    f = factorial(n); S = 0
    for k in (0..n):
        for p in Partitions(n, max_part=k, inner=[k], length=n+1-k):
            S += (f // p.aut())
    return S
print([A121726(n) for n in (1..23)]) # Peter Luschny, Nov 20 2020

a(n) = A006231(n) + 1 = A002104(n) - (n-1). - Franklin T. Adams-Watters, Aug 29 2006

E.g.f.: exp(x)*(log(1/(1-x)) - x + 1). - Geoffrey Critzer, Nov 07 2015

More terms from Franklin T. Adams-Watters, Aug 29 2006

More terms from R. J. Mathar, Sep 02 2006

A305407 Expansion of e.g.f. 1/(1 + log(1 - x)*exp(x)).

Original entry on oeis.org

1, 1, 5, 32, 274, 2939, 37833, 568210, 9753280, 188342949, 4041170695, 95380234366, 2455830637412, 68501591450447, 2057726452045145, 66227424015265178, 2273614433910697920, 82932491842062712873, 3202994529476330549163, 130577628147690206429038, 5603479009890212632226756

Offset: 0

Ilya Gutkovskiy, May 31 2018

nonn

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 32*x^3/3! + 274*x^4/4! + 2939*x^5/5! + 37833*x^6/6! + ...

Seiichi Manyama, Table of n, a(n) for n = 0..399
Index entries for sequences related to logarithmic numbers

Cf. A002104, A006153, A007840, A009324, A191365.

a:=series(1/(1+log(1-x)*exp(x)),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019

nmax = 20; CoefficientList[Series[1/(1 + Log[1 - x] Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[HypergeometricPFQ[{1, 1, 1 - k}, {2}, -1] a[n - k]/(k - 1)!, {k, 1, n}]; Table[n! a[n], {n, 0, 20}]

a(n) ~ n! / ((1 + exp(r)/r) * (1 - exp(-r))^(n+1)), where r = 0.62747017959751658496114808922921433658821962606026068561095... is the root of the equation r*exp(1 - exp(-r)) = 1. - Vaclav Kotesovec, Mar 26 2019

a(0) = 1; a(n) = Sum_{k=1..n} A002104(k) * binomial(n,k) * a(n-k). - Seiichi Manyama, May 04 2022

A339016 A classification of permutations based on their cycle length and the size of the centralizer of their cycle type. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 3, 21, 0, 0, 0, 0, 35, 85, 0, 0, 0, 0, 55, 255, 410, 0, 0, 0, 0, 0, 1015, 1659, 2366, 0, 0, 0, 0, 0, 2485, 10528, 11242, 16065, 0, 0, 0, 0, 0, 2240, 58149, 92064, 84762, 125665, 0, 0, 0, 0, 0, 0, 228221, 760725, 805530, 722250, 1112074

Offset: 0

Peter Luschny, Nov 19 2020

The size of the centralizer of a partition p is aut(p) = Product_{j = 1..k} m(j)!*j^m(j), where m(j) is the multiplicity of j as a part of p. (For instance p = [2, 2, 2] -> aut(p) = 3!*2^3.)
Let M be the matrix with M(k, r) = Sum_{p in P(n, k)} n! / aut(p) where P(n, k) are the partitions of n with largest part k and length(p) = r. Then T(n, k) = Sum_{j=0..k} M(j, k-j+1), which are the antidiagonal sums of the upper triangular part of the matrix M.
In the example section below it is explained how the matrix M leads to a two-dimensional classification of the permutations of [n] which project to the unsigned Stirling cycle numbers and the number of permutations with longest cycle length.

			Triangle starts:
0:  [1]
1:  [0, 1]
2:  [0, 0, 2]
3:  [0, 0, 0, 6]
4:  [0, 0, 0, 3,  21]
5:  [0, 0, 0, 0,  35,   85]
6:  [0, 0, 0, 0,  55,  255,     410]
7:  [0, 0, 0, 0,   0, 1015,    1659,  2366]
8:  [0, 0, 0, 0,   0, 2485,   10528, 11242, 16065]
9:  [0, 0, 0, 0,   0, 2240,   58149, 92064, 84762, 125665]
----------------------------------------------------------
Sum  1, 1, 2, 9, 111, 6080, 2331767, ...
.
Examples for the basic two-dimensional classification of permutations (dots indicate zeros):
.
* Case n = 6:
   |   1     2     3    4    5    6  | Sum
-------------------------------------|----
1  |   .     .     .    .    .   [1] |   1
2  |   .     .   [ 15] [45] [15]     |  75
3  |   .   [ 40] [120] [40]          | 200
4  |   .   [ 90] [ 90]               | 180
5  |   .   [144]                     | 144
6  | [120]                           | 120
-------------------------------------|----
Sum| 120,  274,   225,  85,  15,  1  | 720
.
Antidiagonals: [40 + 15, 90 + 120 + 45, 120 + 144 + 90 + 40 + 15 + 1]
Leads to row 6 (disregarding leading zeros): 55 + 255 + 410 = 720.
.
* Case n = 7:
   |  1      2     3     4     5    6    7  | Sum
--------------------------------------------|-----
1  |  .      .     .     .     .    .   [1] |    1
2  |  .      .     .   [105] [105] [21]     |  231
3  |  .      .   [490] [420] [ 70]          |  980
4  |  .    [420] [630] [210]                | 1260
5  |  .    [504] [504]                      | 1008
6  |  .    [840]                            |  840
7  | [720]                                  |  720
--------------------------------------------|-----
Sum| 720,  1764,  1624, 735,  175,  21,  1  | 5040
.
Antidiagonals: [420+490+105, 504+630+420+105, 720+840+504+210+70+21+1]
Leads to row 7 (disregarding leading zeros): 1015 + 1659 + 2366 = 5040
.
* Column sums of the matrix give the unsigned Stirling cycle numbers, A132393.
* Row sums of the matrix give the number of permutations of n elements whose longest cycle have length k, A126074.
* The main antidiagonal of the matrix gives the number of n-permutations that are pure cycles of length n - k, A092271.
* The entries of the matrix sum to n!. In particular the sum over all row sums, the sum over all column sums, and the sum over all antidiagonal sums is n!.
* The columns of the triangle are finite in the sense that their entries become ultimately zero. Column sums of the triangle are A339015.

Cf. A000142 (row sums), A339015 (column sums), A132393, A126074, A092271, A121726, A339033, A006231, A002104.

# For illustration computes also A132393 and A126074 (remove the #).
def A339016Row(n):
    f = factorial(n); M = matrix(n + 2)
    for k in (0..n):
        for p in Partitions(n, max_part=k, inner=[k]):
            M[k, len(p)] += (f // p.aut())
    # print("max cyc len", [sum(M[k, j] for j in (0..n+1)) for k in (0..n)])
    # print("Stirling 1 ", [sum(M[j, k] for j in (0..n+1)) for k in (0..n)])
    if n == 0: return [1]
    return [sum(M[j, k-j+1] for j in srange(k, 0, -1)) for k in (0..n)]
for n in (0..9): print(A339016Row(n))

T(n, n) = A006231(n) + 1 = A002104(n) - (n-1) (after Franklin T. Adams-Watters in A121726).

A291484 Expansion of e.g.f. arctanh(x)*exp(x).

Original entry on oeis.org

0, 1, 2, 5, 12, 49, 190, 1301, 7224, 69441, 495898, 6095429, 53005700, 792143793, 8110146070, 142633278997, 1679413757168, 33964965659649, 451969255722162, 10331348137881349, 153288815339260796, 3907452790559751857, 63949589015139119598, 1798373345567005989781, 32179694275204166066728

Offset: 0

Ilya Gutkovskiy, Aug 24 2017

nonn

			E.g.f.: A(x) = x/1! + 2*x^2/2! + 5*x^3/3! + 12*x^4/4! + 49*x^5/5! + ...

Cf. A002104, A002741, A009739, A009832, A010050, A012709, A087208 (first differences), A279927.

a:=series(arctanh(x)*exp(x),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 27 2019

nmax = 24; Range[0, nmax]! CoefficientList[Series[ArcTanh[x] Exp[x], {x, 0, nmax}], x]
nmax = 24; Range[0, nmax]! CoefficientList[Series[Log[(1 + x)/(1 - x)] Exp[x]/2, {x, 0, nmax}], x]
nmax = 24; Range[0, nmax]! CoefficientList[Series[Sum[x^(2 k + 1)/(2 k + 1), {k, 0, Infinity}] Exp[x], {x, 0, nmax}], x]
Table[Sum[Binomial[n+1,2k+1](n-2k)/(n+1) (2 k)!, {k,0,n/2}],{n,0,12}] (* Emanuele Munarini, Dec 16 2017 *)

makelist(sum(binomial(n+1,2*k+1)*(n-2*k)/(n+1)*(2*k)!,k,0,floor(n/2)),n,0,12); /* Emanuele Munarini, Dec 16 2017 */

first(n) = x='x+O('x^n); Vec(serlaplace(atanh(x)*exp(x)), -n) \\ Iain Fox, Dec 16 2017

E.g.f.: log((1 + x)/(1 - x))*exp(x)/2.
From Emanuele Munarini, Dec 16 2017: (Start)
a(n) = Sum_{k=0..n/2} binomial(n+1,2*k+1)*((n-2*k)/(n+1))*(2*k)!.
a(n+3) - a(n+2) - (n+1)*(n+2)*a(n+1) + (n+1)*(n+2)*a(n) = 1.
a(n+4) - 2*a(n+3) - (n^2+5*n+5)*a(n+2) + 2*(n+2)^2*a(n+1) - (n+1)*(n+2)*a(n) = 0.
(End)
a(n) ~ (n-1)! * (exp(1) - (-1)^n * exp(-1))/2. - Vaclav Kotesovec, Dec 16 2017

A336291 a(n) = (n!)^2 * Sum_{k=1..n} 1 / (k * ((n-k)!)^2).

Original entry on oeis.org

0, 1, 6, 39, 424, 7905, 227766, 9324511, 512970144, 36452217969, 3247711402870, 354391640998791, 46474986465907176, 7210874466760191409, 1306387103147257800774, 273269900360634449732895, 65363179181419926246184576, 17726298367452515070739268001

Offset: 0

Ilya Gutkovskiy, Jul 16 2020

nonn

Cf. A002104, A002720, A006040, A336292.

Table[(n!)^2 Sum[1/(k ((n - k)!)^2), {k, 1, n}], {n, 0, 17}]
nmax = 17; CoefficientList[Series[-Log[1 - x] BesselI[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2

a(n) = (n!)^2 * sum(k=1, n, 1 / (k * ((n-k)!)^2)); \\ Michel Marcus, Jul 17 2020

Sum_{n>=0} a(n) * x^n / (n!)^2 = -log(1 - x) * BesselI(0,2*sqrt(x)).
a(n) ~ BesselI(0,2) * (n!)^2 / n. - Vaclav Kotesovec, Jul 17 2020
a(n) = Sum_{k=1..n} (k-1)!*k!*binomial(n,k)^2. - Ridouane Oudra, Jun 15 2025

A356925 E.g.f. satisfies A(x) = 1/(1 - x)^(exp(x) * A(x)).

Original entry on oeis.org

1, 1, 6, 51, 614, 9655, 188209, 4389532, 119363488, 3711190881, 129932611723, 5060364817200, 217054300138136, 10168837756846145, 516709033266165479, 28306732060349788908, 1663231006737554997168, 104344911495734048046929

Offset: 0

Seiichi Manyama, Sep 04 2022

nonn

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

Cf. A191365, A356926.
Cf. A356752, A356753.
Cf. A000272, A002104, A356927.

nmax = 20; CoefficientList[Series[LambertW[E^x * Log[1-x]]/(E^x * Log[1-x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 14 2022 *)

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(exp(x)*log(1-x)))))

my(N=20, x='x+O('x^N)); Vec(serlaplace(lambertw(exp(x)*log(1-x))/(exp(x)*log(1-x))))

E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-exp(x) * log(1-x))^k / k!.
E.g.f.: A(x) = exp( -LambertW(exp(x) * log(1-x)) ).
E.g.f.: A(x) = LambertW(exp(x) * log(1-x))/(exp(x) * log(1-x)).
a(n) ~ sqrt(1 + exp(1+r)/(1-r)) * n^(n-1) / (r^(n - 1/2) * exp(n-1)), where r = 0.249272970940807862774650581662931601739615720771408527... is the root of the equation exp(1+r) * log(1-r) = -1. - Vaclav Kotesovec, Nov 14 2022

A114633 a(n) = (n+1)(n+2)/2 Sum_{k=0..floor(n/2)} n!/(n-2*k)!.

Original entry on oeis.org

1, 3, 18, 70, 555, 2961, 31108, 213228, 2799765, 23455135, 369569046, 3659001138, 67261566463, 768390239085, 16142775951240, 209002145031256, 4939689441079593, 71478733600689723, 1877081987610245530, 30021068112289683870, 867211878275933435091, 15190660464818580038473

Offset: 0

Creighton Dement, Feb 17 2006

nonn

Formula was found by Paul D. Hanna.

Related to logarithmic numbers A002104.

Vincenzo Librandi, Table of n, a(n) for n = 0..450

Cf. A002104, A087208.

a:= n-> (n+1)*(n+2)/2*sum(n!/(n-2*k)!,k=0..floor(n/2)): seq(a(n), n=0..20);

Rest[Rest[With[{nn=25}, CoefficientList[Series[Exp[x]/(1 - x^2)(x^2/2), {x, 0, nn}], x] Range[0, nn]!]]] (* Vincenzo Librandi, Sep 03 2017 *)

a(n) = A087208(n)*(n+1)*(n+2)/2. - Paul D. Hanna

E.g.f.: exp(x)/(1-x^2)*(x^2/2) (with offset 2). - Zerinvary Lajos, Apr 03 2009

More terms from Vincenzo Librandi, Sep 03 2017

A302581 a(n) = n! * [x^n] -exp(-nx)log(1 - x).

Original entry on oeis.org

0, 1, -3, 20, -186, 2249, -33360, 586172, -11901008, 274098393, -7060189120, 201092672604, -6275340884736, 212915635727313, -7803567334571008, 307245946117223700, -12933084380738398208, 579587518114690731601, -27550568677612746940416, 1384553892443352890245636

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

sign

N. J. A. Sloane, Transforms
Index entries for sequences related to logarithmic numbers

Cf. A002104, A002741, A104150, A134095, A190314.

Table[n! SeriesCoefficient[-Exp[-n x] Log[1 - x], {x, 0, n}], {n, 0, 19}]
Table[Sum[(-n)^(n - k) (k - 1)! Binomial[n, k], {k, 1, n}], {n, 0, 19}]
nmax = 20; CoefficientList[Series[-Log[1 - LambertW[x]]/(1 + LambertW[x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 09 2019 *)

a(n) = Sum_{k=1..n} (-n)^(n-k)*(k-1)!*binomial(n,k).
E.g.f.: -log(1 - LambertW(x))/(1 + LambertW(x)). - Vaclav Kotesovec, Jun 09 2019
a(n) ~ -(-1)^n * log(2) * n^n. - Vaclav Kotesovec, Jun 09 2019

cp's OEIS Frontend

A353548 Expansion of e.g.f. -log(1-4*x) * exp(x)/4.

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A353549 Expansion of e.g.f. log(1+3*x) * exp(x)/3.

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A121726 Sum sequence A000522 then subtract 0,1,2,3,4,5,...

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A305407 Expansion of e.g.f. 1/(1 + log(1 - x)*exp(x)).

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A339016 A classification of permutations based on their cycle length and the size of the centralizer of their cycle type. Triangle read by rows, T(n, k) for 0 <= k <= n.

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A291484 Expansion of e.g.f. arctanh(x)*exp(x).

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A336291 a(n) = (n!)^2 * Sum_{k=1..n} 1 / (k * ((n-k)!)^2).

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A356925 E.g.f. satisfies A(x) = 1/(1 - x)^(exp(x) * A(x)).

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A353548 Expansion of e.g.f. -log(1-4x) exp(x)/4.

A353549 Expansion of e.g.f. log(1+3x) exp(x)/3.

A114633 a(n) = (n+1)(n+2)/2 Sum_{k=0..floor(n/2)} n!/(n-2*k)!.

A302581 a(n) = n! * [x^n] -exp(-nx)log(1 - x).