A320350 -id:A320350 - cp's OEIS frontend

A185393 Decimal expansion of e/(e-1) = 1 + 1/e + 1/e^2 + ...

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

Offset: 1

Charles R Greathouse IV, Mar 20 2012

			1.58197670686932642438500200510901155854686930107539613626678705964804...

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année, MP, Dunod, 1997, Exercice 3.3.29.a) pp. 286 and 307.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Tannery's theorem.
Index entries for transcendental numbers

Cf. A254445, A254446, A320349, A320350.

Apart from 1st digit the same as A073333.

RealDigits[E/(E - 1), 10, 100][[1]] (* G. C. Greubel, Jun 29 2017 *)

PARI
```
exp(1)/(exp(1)-1)
```

from sympy import E
print(list(map(int, str((E/(E-1)).n(88))[:-1].replace(".", "")))) # Michael S. Branicky, May 25 2022

Equals Sum_{n>=0} 1/exp(n). - Vaclav Kotesovec, Jan 30 2015
From Vaclav Kotesovec, Oct 13 2018: (Start)
Equals 1 - LambertW(exp(1/(1 - exp(1))) / (1 - exp(1))).
Equals -LambertW(-1, exp(1/(1 - exp(1))) / (1 - exp(1))).
(End)
Equals Sum_{k>=0} (-1)^k*B(k)/k!, where B(k) is the k-th Bernoulli number. - Amiram Eldar, May 08 2021
Equals Integral_{x=0..oo} exp(-floor(x)) dx (Monier). - Bernard Schott, May 08 2022
Equals lim_{n->oo} Sum_{k=1..n} (k/n)^n (via Tannery's theorem). - Stoyan Apostolov, May 24 2022

A305550 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k).

Original entry on oeis.org

1, 1, 3, 19, 135, 1171, 12543, 156619, 2185095, 33787171, 579341583, 10927420219, 223956672855, 4940901389971, 116678668726623, 2938719256363819, 78709685812037415, 2234633592020685571, 67005923560416063663, 2114549937496479803419, 70024572874029038582775, 2427790107567416812409971

Offset: 0

Ilya Gutkovskiy, Jun 15 2018

nonn

Stirling transform of A088311.
From Peter Bala, Jul 08 2022: (Start)
Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 16 we obtain the sequence [1, 1, 3, 3, 7, 3, 15, 11, 7, 3, 15, 11, 7, 3, 15, 11, ...], with an apparent period of 4 beginning at a(4). Cf. A167137.
More generally, we conjecture that the same property holds for integer sequences having an e.g.f. of the form G(exp(x) - 1), where G(x) is an integral power series. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..400
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000009, A088311, A167137, A306045, A320350.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> add(Stirling2(n, k)*k!*b(k), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018

nmax = 21; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[(-1)^k (Exp[x] - 1)^k/(k ((Exp[x] - 1)^k - 1)), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] PartitionsQ[k] k!, {k, 0, n}], {n, 0, 21}]

E.g.f.: exp(Sum_{k>=1} (-1)^k*(exp(x) - 1)^k/(k*((exp(x) - 1)^k - 1))).
a(n) = Sum_{k=0..n} Stirling2(n,k)*A088311(k).
From Vaclav Kotesovec, Jun 17 2018: (Start)
a(n) ~ n! * exp(Pi*sqrt(n/(6*log(2))) + (1/log(2) - 1) * Pi^2/48) / (2^(9/4) * 3^(1/4) * n^(3/4) * (log(2))^(n + 1/4)).
a(n) ~ sqrt(Pi) * exp(Pi*sqrt(n/(6*log(2))) + (1/log(2) - 1) * Pi^2/48 - n) * n^(n + 1/2) / (2^(7/4) * 3^(1/4) * n^(3/4) * (log(2))^(n + 1/4)).
(End)

A320349 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1/(1 - x))^k).

Original entry on oeis.org

1, 1, 5, 32, 278, 2894, 35986, 514128, 8306448, 149558688, 2968216944, 64314676128, 1510065781968, 38178537908016, 1033794746169168, 29840453678758272, 914461132860063360, 29645845798652997120, 1013511411165693991680, 36436289007997132646400, 1373976152501162688288000

Offset: 0

Ilya Gutkovskiy, Oct 11 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..413

Cf. A000041, A000203, A048994, A053529, A167137, A306042, A320350.

seq(n!*coeff(series(mul(1/(1-log(1/(1-x))^k),k=1..100),x=0,21),x,n),n=0..20); # Paolo P. Lava, Jan 09 2019

nmax = 20; CoefficientList[Series[Product[1/(1 - Log[1/(1 - x)]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] Log[1/(1 - x)]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] PartitionsP[k] k!, {k, 0, n}], {n, 0, 20}]

E.g.f.: exp(Sum_{k>=1} sigma(k)*log(1/(1 - x))^k/k).
a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A000041(k)*k!.
From Vaclav Kotesovec, Oct 13 2018: (Start)
a(n) ~ n! * exp(n + Pi*sqrt(2*n/(3*(exp(1) - 1))) + Pi^2/(12*(exp(1) - 1))) / (4 * sqrt(3) * n * (exp(1) - 1)^n).
a(n) ~ sqrt(Pi) * exp(Pi*sqrt(2*n/(3*(exp(1) - 1))) + Pi^2/(12*(exp(1) - 1))) * n^(n - 1/2) / (2^(3/2) * sqrt(3) * (exp(1) - 1)^n).
(End)

A298905 Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k).

Original entry on oeis.org

1, 1, 1, 8, -8, 224, -712, 9120, -53496, 980088, -14394648, 264140832, -4113747024, 59028225840, -545558201424, -4191307074432, 450100910950272, -17302659472138752, 530508727766191104, -14790496500550616832, 408513443917280375808, -12274212131738107257600

Offset: 0

Ilya Gutkovskiy, Jun 18 2018

sign

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000009, A088311, A305550, A306023, A306042, A320350.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> add(Stirling1(n, j)*b(j)*j!, j=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 18 2018

nmax = 21; CoefficientList[Series[Product[(1 + Log[1 + x]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) Log[1 + x]^k/(k (1 - Log[1 + x]^k)), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] PartitionsQ[k] k!, {k, 0, n}], {n, 0, 21}]

E.g.f.: exp(Sum_{k>=1} (-1)^(k+1)*log(1 + x)^k/(k*(1 - log(1 + x)^k))).

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000009(k)*k!.

A307524 Expansion of e.g.f. Product_{k>=1} (1 + log(1/(1 - x))^k)/(1 - log(1/(1 - x))^k).

Original entry on oeis.org

1, 2, 10, 76, 724, 8368, 113792, 1771824, 31001424, 601677888, 12818974848, 297223165248, 7446226027584, 200354793323904, 5760239869401984, 176170480317568512, 5709535272618925824, 195419487662892221184, 7042458625343222876928, 266500916470984705887744

Offset: 0

Ilya Gutkovskiy, Apr 12 2019

nonn

Exponential convolution of A320349 and A320350.

Cf. A015128, A306045, A307523, A320349, A320350.

nmax = 19; CoefficientList[Series[Product[(1 + Log[1/(1 - x)]^k)/(1 - Log[1/(1 - x)]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2 k] - DivisorSigma[1, k]) Log[1/(1 - x)]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/EllipticTheta[4, 0, Log[1/(1 - x)]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] Sum[PartitionsQ[j] PartitionsP[k - j], {j, 0, k}] k!, {k, 0, n}], {n, 0, 19}]

E.g.f.: exp(Sum_{k>=1} (sigma(2*k) - sigma(k))*log(1/(1 - x))^k/k).
E.g.f.: 1/theta_4(log(1/(1 - x))).
a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A015128(k)*k!.
a(n) ~ sqrt(Pi) * exp(Pi*sqrt(n/(exp(1)-1)) + Pi^2/(8*(exp(1)-1))) * n^(n - 1/2) / (2^(5/2) * (exp(1)-1)^n). - Vaclav Kotesovec, Apr 13 2019

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A185393 Decimal expansion of e/(e-1) = 1 + 1/e + 1/e^2 + ...

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A305550 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k).

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A320349 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1/(1 - x))^k).

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A298905 Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k).

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Maple

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A307524 Expansion of e.g.f. Product_{k>=1} (1 + log(1/(1 - x))^k)/(1 - log(1/(1 - x))^k).

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