A177885 -id:A177885 - cp's OEIS frontend

A383985 Series expansion of the exponential generating function LambertW(1-exp(x)), see A000169.

0, 1, -1, 4, -23, 181, -1812, 22037, -315569, 5201602, -97009833, 2019669961, -46432870222, 1168383075471, -31939474693297, 942565598033196, -29866348653695203, 1011335905644178273, -36446897413531401020, 1392821757824071815641, -56259101478392975833333

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..200
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, permutative commutative operad "Perm/Com2".

Cf. A002050, A006531, A084099, A101851, A114285, A177885, A225883, A383986, A383987, A383988, A383989.

Composition of A000169 with signs and 1-exp(x).

nn = 20; f[x_] := -Sum[k^(k - 1)*(1 - Exp[x])^k/k!, {k, nn}];
Range[0, nn]! * CoefficientList[Series[f[x], {x, 0, nn}], x]

A383990 Series expansion of the exponential generating function exp(-dend(-x))-1 where dend(x) = (1 - sqrt(1+4x)) / (2x) + 1 (given by A000108).

Original entry on oeis.org

0, 1, -3, 19, -191, 2661, -47579, 1040047, -26888511, 802727209, -27178685459, 1029077910411, -43086906080063, 1976633329627789, -98597207392040811, 5313105048925173991, -307587436319162110079, 19038773384213189214417, -1254686724727364725716131

Offset: 0

Michael De Vlieger, May 16 2025

sign

The series -dend(-x) is the inverse for the substitution of the series dias(x), given by the suspension of the Koszul dual of dias. - Bérénice Delcroix-Oger, May 28 2025

Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3, diassociative operad "Dias".

Cf. A003725, A006531, A097388, A111884, A112242, A177885, A318215, A383991, A383992, A383993, A383994, A383995. Composition of exp(x)-1 with -A000108(-x).

A230284 Denominators to Dirichlet inverse of Euler totient based version of series expansion for x/LambertW(x).

Original entry on oeis.org

1, 1, 2, 3, 8, 15, 144, 35, 5760, 315, 5600, 693, 43545600, 1001, 6706022400, 6435, 14014, 109395, 376610217984000, 46189, 128047474114560000, 323323, 2540395, 2028117, 26976017466662584320000, 96577, 3241475864250624, 35102025, 2126818694000, 5386025

Offset: 1

Mats Granvik, Oct 15 2013

G. C. Greubel, Table of n, a(n) for n = 1..385

Cf. A191898, A177885, A230283 (numerators).

Similar to but strictly different from A264235.

Clear[nn, n, k, s, x]; nn = 22; Denominator[CoefficientList[1 + Integrate[1 + Expand[Sum[Exp[Limit[Zeta[s]*Sum[(If[n == 1, 0, Table[DivisorSum[m, # MoebiusMu[#] &], {m, nn}][[GCD[n, k]]]])/(k)^(s - 1), {k, 1, n}], s -> 1]]*(-x)^n, {n, 1, nn}]], x], x]]

A230283 Numerators to Dirichlet inverse of Euler totient based version of series expansion for x/LambertW(x).

Original entry on oeis.org

1, 1, -1, 2, -9, 8, -625, 2, -117649, 128, -6561, 8, -25937424601, 18, -23298085122481, 16, -9, 32768, -48661191875666868481, 400, -104127350297911241532841, 648, -81, 256, -907846434775996175406740561329, 490, -59604644775390625, 1024, -2541865828329, 1296

Offset: 1

Mats Granvik, Oct 15 2013

The coefficients of the series expansion of x/Lambert(x) expanded at 0 can be seen as exponentiated numerators in convergents of zeta function limits of truncated Dirichlet series for logarithms. Those numerators are defined by simple recurrences. Letting those recurrences run in cross directions to each other, one get the Dirichlet inverse of the Euler totient in a greatest common divisor matrix, and the von Mangoldt function as convergents of Dirichlet series. Since x/LambertW(x) is good at approximately describing the nontrivial Riemann zeta zeros and since the Riemann zeta zeros are the frequencies that build up the von Mangoldt function, this prime number or von Mangoldt function version of the x/LambertW(x) is motivated.

G. C. Greubel, Table of n, a(n) for n = 1..385

Cf. A191898, A177885, A230284 (denominators).

Clear[nn, n, k, s, x]; nn = 22; Numerator[CoefficientList[1 + Integrate[1 + Expand[Sum[Exp[Limit[Zeta[s]*Sum[(If[n == 1, 0, Table[DivisorSum[m, # MoebiusMu[#] &], {m, nn}][[GCD[n, k]]]])/(k)^(s - 1), {k, 1, n}], s -> 1]]*(-x)^n, {n, 1, nn}]], x], x]]

A274377 E.g.f. satisfies: A(x)^A(x) = exp(2x) A(-x)^A(-x).

Original entry on oeis.org

1, 1, 0, 1, 0, 16, 0, 736, 0, 67096, 0, 10163176, 0, 2306198896, 0, 732199108096, 0, 309860700130816, 0, 168568765338224896, 0, 114619705107961862656, 0, 95251358122177791486976, 0, 94984793274454431691503616, 0, 111939507886837612683516276736, 0, 153907136552991217284274400567296, 0, 244164979570216285201628515234840576, 0, 442692827509235885935744380253757341696, 0, 909667081143908558901949811564629988048896

Offset: 0

Paul D. Hanna, Aug 23 2016

nonn

a(2*n+1) = 6 (mod 10) for n>1 (conjecture).

			E.g.f.: A(x) = 1 + x + x^3/3! + 16*x^5/5! + 736*x^7/7! + 67096*x^9/9! + 10163176*x^11/11! + 2306198896*x^13/13! + 732199108096*x^15/15! + 309860700130816*x^17/17! + 168568765338224896*x^19/19! +...
such that A(x)^A(x) / A(-x)^A(-x) = exp(2*x).
RELATED SERIES.
A(x)^A(x) = 1 + x + 2*x^2/2! + 4*x^3/3! + 16*x^4/4! + 56*x^5/5! + 426*x^6/6! + 2262*x^7/7! + 26944*x^8/8! + 191536*x^9/9! + 3126160*x^10/10! +...+ A275764(n)*x^n/n! +...
Series_Reversion(A(x) - 1) = x - x^3/6 - x^5/20 - x^7/42 - x^9/72 - x^11/110 - x^13/156 - x^15/210 - x^17/272 +...+ -x^(2*n+1)/(2*n*(2*n+1)) +...
Also,
Series_Reversion(A(x) - 1) = (G(x) - G(-x))/2, where G(x) = (1+x)*log(1+x) = Series_Reversion(x/LambertW(x) - 1), and begins:
G(x) = x + x^2/2 - x^3/6 + x^4/12 - x^5/20 + x^6/30 - x^7/42 + x^8/56 - x^9/72 + x^10/90 - x^11/110 + x^12/132 +...+ (-x)^n/(n*(n-1)) +...
GENERATING METHOD.
Start with a(0)=1, a(1)=1, and set a(2*n)=0 for n>0, then use the following criterion to determine the odd-indexed terms.
Given partial sum A(x,2*n) = Sum_{k=0..2*n} a(k)*x^k/k!, and sufficiently large N, the odd-indexed term a(2*n+1) satisfies:
if t > a(2*n+1)/(2*n+1)!, then
t > [x^(2*n+1)] ( A(x,2*n) +  t*x^(2*n+1) )^(1-1/N)
else if t <= a(2*n+1)/(2*n+1)! , then
t < [x^(2*n+1)] ( A(x,2*n) + t*x^(2*n+1) )^(1-1/N);
this criterion defines each term of this sequence for n>1.
Using the same method as above, but without setting even-indexed terms to zero, generates x/LambertW(x), e.g.f. of A177885.
RELATED SERIES.
log(A(x)) = x - x^2/2! + 3*x^3/3! - 10*x^4/4! + 60*x^5/5! - 346*x^6/6! + 3108*x^7/7! - 25600*x^8/8! + 306120*x^9/9! - 3283696*x^10/10! + 49021368*x^11/11! - 648526000*x^12/12! + 11606584080*x^13/13! - 182697457216*x^14/14! +...

Paul D. Hanna, Table of n, a(n) for n = 0..401

Cf. A177885, A275759, A275764.

{a(n) = my(A = 1 + serreverse(x - sum(m=1,n\2+1, x^(2*m+1)/(2*m*(2*m+1)) ) +x^2*O(x^n) ) ); n!*polcoeff(A,n)}
for(n=0,40,print1(a(n),", "))

/* Generating method (using sufficiently large N and precision) */
\p100
{a(n) = my(N=10^(3*n), A=[1,1]); for(i=0,n\2, A=concat(A,[0,0]); A[#A] = round( (#A-1)!*polcoeff( N*1.* Ser(A)^(1-1/N), #A-1) )/(#A-1)! ); n!*A[n+1]}
for(n=0,40,print1(a(n),", "))

E.g.f.: 1 + Series_Reversion( log( sqrt( (1+x)^(1+x) / (1-x)^(1-x) ) ) ).
E.g.f.: 1 + Series_Reversion( (G(x) - G(-x))/2 ), where G(x) = Series_Reversion(x/LambertW(x) - 1) = (1+x)*log(1+x).
E.g.f.: 1 + Series_Reversion( x - Sum_{n>=1} x^(2*n+1)/(2*n*(2*n+1)) ).
If n is odd then a(n) ~ c * d^n * n^(n-1) / exp(n), where d = 1.37441749603820461..., c = 0.6508250221842049... . - Vaclav Kotesovec, Sep 22 2016

A362569 E.g.f. satisfies A(x) = exp(x/A(x)^(x^2)).

Original entry on oeis.org

1, 1, 1, 1, -23, -119, -359, 6721, 78961, 450577, -7867439, -160506719, -1421049959, 23995634521, 745945175977, 9197488067041, -152057966904479, -6667968305775839, -107047941299543519, 1740437689443523777, 102311231044267813321, 2043217889363061489961

Offset: 0

Seiichi Manyama, Apr 25 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 0..427
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A177885, A362568.

Cf. A362571.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(-lambertw(x^3)))))

E.g.f.: (x^3 / LambertW(x^3))^(1/x^2) = exp(LambertW(x^3) / x^2) = exp(x * exp(-LambertW(x^3))).
a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k * (n-2*k)^k * binomial(n-2*k-1,k)/(n-2*k)!.
E.g.f.: Sum_{k>=0} (-k*x^2 + 1)^(k-1) * x^k / k!.

A305787 Inverse Euler transform of (-n)^n.

Original entry on oeis.org

-1, 4, -23, 223, -2800, 42599, -763220, 15734388, -366715248, 9533820200, -273549419552, 8586984241870, -292755986184548, 10772849584162694, -425587711650564816, 17966217346985801150, -807152054953801845760, 38451365602113718874568, -1936082850634342992601636

Offset: 1

Seiichi Manyama, Jun 10 2018

sign

			(1-x) * (1-x^2)^(-4) * (1-x^3)^23 * (1-x^4)^(-223) * ... =  1 - x + 4*x^2 - 27*x^3 + 256*x^4 - ... .

Seiichi Manyama, Table of n, a(n) for n = 1..386
N. J. A. Sloane, Transforms

Cf. A177885, A305754.

Product_{k>=1} 1/(1-x^k)^{a(k)} = Sum_{n>=0} (-n * x)^n.

a(n) ~ (-1)^n * n^n. - Vaclav Kotesovec, Oct 09 2019

A326501 a(n) = Sum_{k=0..n} (-k)^k.

Original entry on oeis.org

1, 0, 4, -23, 233, -2892, 43764, -779779, 15997437, -371423052, 9628576948, -275683093663, 8640417354593, -294234689237660, 10817772136320356, -427076118244539019, 18019667955465012597, -809220593930871751580, 38537187481365665823844

Offset: 0

Seiichi Manyama, Sep 12 2019

Seiichi Manyama, Table of n, a(n) for n = 0..386

Cf. A001099, A062970, A177885.

a:= proc(n) option remember; `if`(n<0, 0, (-n)^n+a(n-1)) end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 12 2019

RecurrenceTable[{a[0] == 1, a[n] == a[n-1] + (-n)^n}, a, {n, 0, 23}] (* Jean-François Alcover, Nov 27 2020 *)

PARI
```
{a(n) = sum(k=0, n, (-k)^k)}
```

from itertools import accumulate, count, islice
def A326501_gen(): # generator of terms
    yield from accumulate((-k)**k for k in count(0))
A326501_list = list(islice(A326501_gen(),10)) # Chai Wah Wu, Jun 18 2022

a(n) = 1 + (-1)^n * A001099(n).

A362568 E.g.f. satisfies A(x) = exp(x/A(x)^x).

Original entry on oeis.org

1, 1, 1, -5, -23, 121, 1321, -7349, -148175, 853777, 27840241, -163354949, -7934320679, 46820981065, 3203091569497, -18833438286389, -1742847946697759, 10137524365568161, 1230956201929018465, -7042544858204663813, -1095864481054115534519

Offset: 0

Seiichi Manyama, Apr 25 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 0..417
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A177885, A362569.

Cf. A361777.

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

E.g.f.: (x^2 / LambertW(x^2))^(1/x) = exp(LambertW(x^2) / x) = exp(x * exp(-LambertW(x^2))).
a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * (n-k)^k * binomial(n-k-1,k)/(n-k)!.
E.g.f.: Sum_{k>=0} (-k*x + 1)^(k-1) * x^k / k!.

A340473 a(n) = n! [x^n] W(-W(x))/(-W(x)), where W(x) is the Lambert W function.

Original entry on oeis.org

1, 1, 1, 7, 13, 321, 31, 42673, -214983, 12251809, -156239909, 6366130761, -135725103227, 5265915854785, -155145910919817, 6318044844152161, -232403136941014799, 10299509100942804033, -446889500139353805773, 21789892230658085847673, -1078684347590588362463619

Offset: 0

Peter Luschny, Jan 08 2021

sign

Let LW(x) = W(-W(x))/(-W(x)) denote the function in the definition and let T(x) = -W(-x) be Euler's tree function A000169, and L(x) = W(-x)/(-x) the labeled tree function A000272, then LW(x) = L(W(x)), and TW(x) := -T(W(-x)) is A097174, and RW(x) := T(-W(-x)) is A207833.

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

Cf. A097174, A177885, A207833, A227176, A340474.

Cf. A000169, A000272.

W := x -> LambertW(x): gf := W(-W(x))/(-W(x)):
ser := series(gf, x, 24): seq(n!*coeff(ser, x, n), n=0..20);

gf := -ProductLog[-ProductLog[x]]/ProductLog[x];
Range[0, 20]! CoefficientList[Series[gf, {x, 0, 20}], x]

my(x='x+O('x^25)); Vec(serlaplace(lambertw(-lambertw(x))/(-lambertw(x)))) \\ Michel Marcus, Jan 09 2021

cp's OEIS Frontend

A383985 Series expansion of the exponential generating function LambertW(1-exp(x)), see A000169.

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A383990 Series expansion of the exponential generating function exp(-dend(-x))-1 where dend(x) = (1 - sqrt(1+4*x)) / (2*x) + 1 (given by A000108).

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A230284 Denominators to Dirichlet inverse of Euler totient based version of series expansion for x/LambertW(x).

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Mathematica

A230283 Numerators to Dirichlet inverse of Euler totient based version of series expansion for x/LambertW(x).

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Mathematica

A274377 E.g.f. satisfies: A(x)^A(x) = exp(2*x) * A(-x)^A(-x).

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A362569 E.g.f. satisfies A(x) = exp(x/A(x)^(x^2)).

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A305787 Inverse Euler transform of (-n)^n.

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A326501 a(n) = Sum_{k=0..n} (-k)^k.

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A362568 E.g.f. satisfies A(x) = exp(x/A(x)^x).

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A383990 Series expansion of the exponential generating function exp(-dend(-x))-1 where dend(x) = (1 - sqrt(1+4x)) / (2x) + 1 (given by A000108).

A274377 E.g.f. satisfies: A(x)^A(x) = exp(2x) A(-x)^A(-x).