A009306 -id:A009306 - cp's OEIS frontend

A302397 Expansion of e.g.f. 1/(1 + x*exp(x)).

1, -1, 0, 3, -4, -25, 114, 287, -4152, 1647, 192230, -807961, -10164804, 111209111, 454840554, -14657978385, 21202175504, 1988791958879, -15488971798194, -260886468394153, 4872247004699460, 23537372210149959, -1365745577227898350, 4274609859520565663, 364461939727273277016

Offset: 0

Ilya Gutkovskiy, Apr 07 2018

sign

			1/(1 + x*exp(x)) = 1 - x/1! + 3*x^3/3! - 4*x^4/4! - 25*x^5/5! + 114*x^6/6! + 287*x^7/7! - 4152*x^8/8! + 1647*x^9/9! + ...

Seiichi Manyama, Table of n, a(n) for n = 0..474

Cf. A006153, A009306, A089148.

a:=series(1/(1+x*exp(x)),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 26 2019

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

E.g.f.: 1/(1 + x*exp(x)).
a(n) = n!*Sum_{k=0..n} (-1)^(n-k)*(n-k)^k/k!.
a(n) = Sum_{k=0..n} (-1)^k*k!*k^(n-k)*binomial(n,k).

A009444 E.g.f. log(1 + x*exp(-x)).

Original entry on oeis.org

0, 1, -3, 11, -58, 409, -3606, 38149, -470856, 6641793, -105398650, 1858413061, -36044759796, 762659322385, -17481598316742, 431535346662645, -11413394655983536, 321989729198400385, -9651573930139850610

Offset: 0

R. H. Hardin

abs(a(n)) is the number of connected functions f:{1,2,...,n}->{1,2,...,n} such that every element is mapped into a recurrent element. Cf. A006153. - Geoffrey Critzer, May 24 2012

G. C. Greubel, Table of n, a(n) for n = 0..400
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 125

Cf. A006153, A009306, A305990.

With[{nmax = 40}, CoefficientList[Series[Log[1 + x*Exp[-x]], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 22 2017 *)

a(n):=(-1)^(n+1)*n!*sum(m^(n-m-1)/(n-m)!,m,1,n); /* Vladimir Kruchinin, Oct 08 2011 */

x='x+O('x^66); /* that many terms */
egf=1/(1+x/exp(x)); /* = 1 - x + 2*x^2 - 7/2*x^3 + 37/6*x^4 - 87/8*x^5 +... */
Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Apr 30 2011 */

A009444 = lambda n: (-1)^(n+1)*factorial(n)*sum(m^(n-m-1)/factorial(n-m) for m in (1..n))
[A009444(n) for n in (0..9)] # Peter Luschny, Jan 18 2016

abs(a(n)) is asymptotic to (n-1)!/LambertW(1)^n. - Vladeta Jovovic, Jul 12 2007
Sequence of absolute values has e.g.f. log(1/(1-x*exp(x))). - Joerg Arndt, Apr 30 2011
a(n) = (-1)^(n+1)*n!*sum(m=1..n, m^(n-m-1)/(n-m)!). - Vladimir Kruchinin, Oct 08 2011
a(n) = (-1)^(n + 1) * n + Sum_{k=1..n-1} (-1)^(n - k) * binomial(n-1,k-1) * (n - k) * a(k). - Ilya Gutkovskiy, Jan 17 2020

Extended with signs by Olivier Gérard, Mar 15 1997

Definition corrected by Joerg Arndt, Apr 30 2011

A238274 Decimal expansion of abs(LambertW(-1)).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Feb 24 2014

			1.37455701074370748653...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.

Cf. A009306, A030178, A177380, A185221.

RealDigits[N[Abs[LambertW[-1]], 105]][[1]]

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

A300452 Logarithmic transform of the cubes A000578.

Original entry on oeis.org

0, 1, 7, 5, -146, -351, 9936, 51421, -1394000, -12844287, 328407400, 4874111901, -115361217696, -2607873466511, 55768370301112, 1866984952934445, -34886452149332864, -1720211491314549375, 26716801597874981064, 1979492625918149729437, -23490293022369696366560, -2777285149336544358953679

Offset: 0

Ilya Gutkovskiy, Mar 06 2018

sign

			E.g.f.: A(x) = x/1! + 7*x^2/2! + 5*x^3/3! - 146*x^4/4! - 351*x^5/5! + 9936*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..408
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Logarithmic Transform
Eric Weisstein's World of Mathematics, Cubic Number

Cf. A000578, A009306, A033464, A279358.

a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n) -add(j*
      binomial(n, j)*t(n-j)*a(j), j=1..n-1)/n))(i->i^3)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 06 2018

nmax = 21; CoefficientList[Series[Log[1 + Exp[x] x (1 + 3 x + x^2)], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: log(1 + exp(x)*x*(1 + 3*x + x^2)).

A346750 Expansion of e.g.f. log( 1 + x^2 * exp(x) / 2 ).

Original entry on oeis.org

0, 0, 1, 3, 3, -20, -135, -189, 3598, 33300, 39105, -2164085, -23831214, -5268042, 3038813869, 36984819795, -59749871880, -8207734934984, -105142191601887, 482549202944307, 37754304692254030, 489494512692093090, -4466445363328684659, -271973408844483808517

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..457

Cf. A000217, A009306, A133189, A300455, A346751, A346752, A346753.

nmax = 23; 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, 23}]

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! * Sum_{k=1..floor(n/2)} (-1)^(k-1) * k^(n-2*k-1)/(2^k * (n-2*k)!). - Seiichi Manyama, Dec 14 2023

A346751 Expansion of e.g.f. log( 1 + x^3 * exp(x) / 3! ).

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 10, -105, -1064, -6076, -16680, 129525, 2642860, 25431406, 130210444, -639438345, -26431524560, -382074099000, -3083015556624, 5641134587049, 726952330301940, 14940678486798610, 173111303303845060, 258953439321230731, -43858702741534022936

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..471

Cf. A000292, A009306, A145453, A346750, A346752, A346754.

nmax = 24; 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, 24}]

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! * Sum_{k=1..floor(n/3)} (-1)^(k-1) * k^(n-3*k-1)/(6^k * (n-3*k)!). - Seiichi Manyama, Dec 14 2023

A346752 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, 35, -504, -6090, -45870, -265155, -990275, 2733731, 113064315, 1571621870, 15859846380, 116145112140, 289646855916, -9965576133855, -255337210989315, -4024508801328785, -47031887951290165, -338016913616223534, 1717029492398463650

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..487

Cf. A000332, A009306, A145454, A346750, A346751, A346755.

nmax = 25; 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, 25}]

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! * Sum_{k=1..floor(n/4)} (-1)^(k-1) * k^(n-4*k-1)/(24^k * (n-4*k)!). - Seiichi Manyama, Dec 14 2023

A300455 Logarithmic transform of the triangular numbers A000217.

Original entry on oeis.org

0, 1, 2, -1, -11, 19, 201, -764, -7426, 52137, 448435, -5377604, -38712486, 777663613, 4258812299, -149524753650, -505685566184, 36733876797025, 30910872539763, -11174584391207360, 25170998506744790, 4101787001153848461, -24862093152821214653, -1776483826032814964966

Offset: 0

Ilya Gutkovskiy, Mar 06 2018

sign

			E.g.f.: A(x) = x/1! + 2*x^2/2! - x^3/3! - 11*x^4/4! + 19*x^5/5! + 201*x^6/6! - 764*x^7/7! - 7426*x^8/8! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..450
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Logarithmic Transform
Eric Weisstein's World of Mathematics, Triangular Number
Index to sequences related to polygonal numbers

Cf. A000217, A009306, A033464, A279361, A300452.

a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n) -add(j*
      binomial(n, j)*t(n-j)*a(j), j=1..n-1)/n))(i->i*(i+1)/2)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 06 2018

nmax = 23; CoefficientList[Series[Log[1 + Exp[x] x (x + 2)/2], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: log(1 + exp(x)*x*(x + 2)/2).

A320939 a(n) = n! * [x^n] log(1 + Sum_{k>=1} k^n*x^k/k!).

Original entry on oeis.org

0, 1, 3, 5, -650, -46071, 3121776, 5538166381, 3146076001776, -10459815889305231, -100694615309371571840, -193538025548431984737219, 38912028315765820944424730112, 2554132880645627969533690819801657, -106074951996903194289368162206783509504

Offset: 0

Ilya Gutkovskiy, Oct 28 2018

sign

a(n) is the n-th term of the logarithmic transform of the n-th powers.

Alois P. Heinz, Table of n, a(n) for n = 0..77
N. J. A. Sloane, Transforms

Cf. A009306, A033464, A279644, A300452.

seq(coeff(series(factorial(n)*log(1+add(k^n*x^k/factorial(k),k=1..n)),x,n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Oct 28 2018

Table[n! SeriesCoefficient[Log[1 + Sum[k^n x^k/k!, {k, 1, n}]], {x, 0, n}], {n, 0, 14}]

cp's OEIS Frontend

A302397 Expansion of e.g.f. 1/(1 + x*exp(x)).

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A009444 E.g.f. log(1 + x*exp(-x)).

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A238274 Decimal expansion of abs(LambertW(-1)).

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

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A300452 Logarithmic transform of the cubes A000578.

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A346750 Expansion of e.g.f. log( 1 + x^2 * exp(x) / 2 ).

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A346751 Expansion of e.g.f. log( 1 + x^3 * exp(x) / 3! ).

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A346752 Expansion of e.g.f. log( 1 + x^4 * exp(x) / 4! ).

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