A320349 -id:A320349 - cp's OEIS frontend

A167137 E.g.f.: P(exp(x)-1) where P(x) is the g.f. of the partition numbers (A000041).

1, 1, 5, 31, 257, 2551, 30065, 407191, 6214577, 105530071, 1972879025, 40213910551, 886979957297, 21044674731991, 534313527291185, 14448883517785111, 414475305054698417, 12568507978358276311, 401658204472560090545, 13490011548122407566871, 474964861088609044357937, 17491333169997896126211031

Offset: 0

Paul D. Hanna, Nov 03 2009

nonn

CONJECTURE: Sum_{n>=0} a(n)^m * log(1+x)^n/n! is an integer series in x for all integer m>0; see A167138 and A167139 for examples.
From Peter Bala, Jul 07 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, 5, 15, 1, 7, 1, 7, 1, 7, ...], with an apparent period of 2 beginning at a(4).
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)

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 31*x^3/3! + 257*x^4/4! +...
A(log(1+x)) = P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A167138, A000041, A019538 (Stirling2), A218482, A305550, A306045, A320349.

Table[Sum[PartitionsP[k]*StirlingS2[n, k]*k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 21 2018 *)
nmax = 20; CoefficientList[Series[1/QPochhammer[E^x - 1], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 22 2021 *)

{a(n)=if(n==0,1,n!*polcoeff(exp(sum(m=1,n,sigma(m)*(exp(x+x*O(x^n))-1)^m/m) ),n))}

{Stirling2(n, k)=if(k<0||k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{a(n)=sum(k=0,n,numbpart(k)*Stirling2(n, k)*k!)}

x='x+O('x^66); Vec( serlaplace( 1/eta(exp(x)-1) ) ) \\ Joerg Arndt, Sep 18 2013

a(n) = Sum_{k=0..n} A000041(k)*Stirling2(n,k)*k! where A000041 is the partition numbers.
E.g.f.: exp( Sum_{n>=1} sigma(n)*[exp(x)-1]^n/n ).
Sum_{n>=0} a(n) * log(1+x)^n/n! = g.f. of the partition numbers (A000041).
Sum_{n>=0} a(n)^2*log(1+x)^n/n! = g.f. of A167138.
From Peter Bala, Sep 18 2013: (Start)
Sum {n >= 0} (-1)^n*a(n)*(log(1 - x))^n/n! = 1 + x + 3*x^2 + 8*x^3 + 21*x^4 + ... is the o.g.f. of A218482.
a(n) is always odd. Congruences for n >= 1: a(2*n) = 2 (mod 3); a(4*n) = 2 (mod 5); a(6*n) = 0 (mod 7); a(10*n) = 7 (mod 11); a(12*n) = 5 (mod 13); a(16*n) = 0 (mod 17). (End)
From Vaclav Kotesovec, Jun 17 2018: (Start)
a(n) ~ n! * exp((1/log(2) - 1) * Pi^2 / 24 + Pi*sqrt(n/(3*log(2)))) / (4 * sqrt(3) * n * (log(2))^n).
a(n) ~ sqrt(Pi) * exp((1/log(2) - 1) * Pi^2 / 24 + Pi*sqrt(n/(3*log(2))) - n) * n^(n + 1/2) / (2^(3/2) * sqrt(3) * n * (log(2))^n). (End)

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

Original entry on oeis.org

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

A306042 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1 + x)^k).

Original entry on oeis.org

1, 1, 3, 8, 50, 94, 2446, -9024, 297216, -3183264, 64191984, -1041792192, 22098943632, -478805234064, 11856288460272, -308662348027008, 8575865689645440, -248582819381690880, 7556655091130023680, -240521346554744194560, 8049494171497089265920, -283469026458500121634560

Offset: 0

Ilya Gutkovskiy, Jun 17 2018

sign

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

Cf. A000041, A006252, A053529, A167137, A320349.

a:=series(mul(1/(1-log(1+x)^k),k=1..100),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019

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

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

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000041(k)*k!.

A320350 Expansion of e.g.f. Product_{k>=1} (1 + log(1/(1 - x))^k).

Original entry on oeis.org

1, 1, 3, 20, 148, 1384, 15728, 207696, 3094152, 51423288, 945943512, 19083180192, 418550811600, 9907493349168, 251588827187280, 6820899616891008, 196645361557479552, 6007407711127690752, 193842462200078260224, 6586904673329133618432, 235079477736802622742528, 8790132360155070084076800

Offset: 0

Ilya Gutkovskiy, Oct 11 2018

nonn

Cf. A000009, A048994, A088311, A298905, A305550, A320349.

seq(n!*coeff(series(mul((1 + log(1/(1 - x))^k),k=1..100),x=0,22),x,n),n=0..21); # Paolo P. Lava, Jan 09 2019

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

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A000009(k)*k!.
From Vaclav Kotesovec, Oct 13 2018: (Start)
a(n) ~ n! * exp(n + Pi*sqrt(n/(3*(exp(1) - 1))) + Pi^2/(24*(exp(1) - 1))) / (4 * 3^(1/4) * n^(3/4) * (exp(1) - 1)^(n + 1/4)).
a(n) ~ sqrt(Pi) * exp(Pi*sqrt(n/(3*(exp(1) - 1))) + Pi^2/(24*(exp(1) - 1))) * n^(n - 1/4) / (2^(3/2) * 3^(1/4) * (exp(1) - 1)^(n + 1/4)).
(End)

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

A308877 Expansion of e.g.f. (1 + log(1 - x))/(1 + 2*log(1 - x)).

Original entry on oeis.org

1, 1, 5, 38, 386, 4904, 74776, 1330272, 27046848, 618653280, 15723024864, 439559609664, 13405656582336, 442915145716224, 15759326934391296, 600783539885546496, 24430204949876794368, 1055516761826050203648, 48286612866726631489536, 2331682676308057000255488

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Cf. A001710, A002866, A008275, A011782, A050351, A088500, A320349, A308878.

nmax = 19; CoefficientList[Series[(1 + Log[1 - x])/(1 + 2 Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[Abs[StirlingS1[n, k]] 2^(k - 1) k!, {k, 1, n}], {n, 1, 19}]]

a(0) = 1; a(n) = Sum_{k=1..n} |Stirling1(n,k)| * 2^(k-1) * k!.

a(n) ~ n! * exp(n/2) / (4 * (exp(1/2) - 1)^(n+1)). - Vaclav Kotesovec, Jun 29 2019

A307525 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1/(1 - x))^k/k!).

Original entry on oeis.org

1, 1, 4, 21, 146, 1240, 12479, 144970, 1908682, 28079550, 456458832, 8125189974, 157190542607, 3284222304545, 73705849847317, 1768479436456975, 45180024672023814, 1224529894981726614, 35096983241255523572, 1060703070504583747430, 33714045363258013414692

Offset: 0

Ilya Gutkovskiy, Apr 12 2019

nonn

Cf. A005651, A140585, A306039, A320349.

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

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} log(1/(1 - x))^(j*k)/(k*(j!)^k)).
a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A005651(k).
a(n) ~ c * sqrt(2*Pi) * n^(n + 1/2) / (exp(1) - 1)^(n+1), where c = A247551 = Product_{k>=2} 1/(1-1/k!). - Vaclav Kotesovec, Apr 13 2019

cp's OEIS Frontend

A167137 E.g.f.: P(exp(x)-1) where P(x) is the g.f. of the partition numbers (A000041).

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

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A306042 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1 + x)^k).

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A320350 Expansion of e.g.f. Product_{k>=1} (1 + log(1/(1 - x))^k).

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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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A308877 Expansion of e.g.f. (1 + log(1 - x))/(1 + 2*log(1 - x)).

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A307525 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1/(1 - x))^k/k!).

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