A167137 -id:A167137 - cp's OEIS frontend

A167138 G.f.: Sum_{n>=0} A167137(n)^2 * log(1+x)^n/n! where Sum_{n>=0} A167137(n)*log(1+x)^n/n! = g.f. of the partition numbers (A000041).

1, 1, 12, 148, 2523, 48996, 1127354, 29348080, 849632392, 27096593838, 943340417806, 35501579861404, 1434531966551084, 61939404662074706, 2844544965703554566, 138338597978951126666, 7098617731036257970895

Offset: 0

Paul D. Hanna, Nov 03 2009

nonn

Conjecture: For all integers m > 0, Sum_{n>=0} L(n)^m * log(1+x)^n/n! is an integer series whenever Sum_{n>=0} L(n)*log(1+x)^n/n! is an integer series.

			G.f.: A(x) = 1 + x + 12*x^2 + 148*x^3 + 2523*x^4 + ...
Illustrate A(x) = Sum_{n>=0} A167137(n)^2*log(1+x)^n/n!:
A(x) = 1 + log(1+x) + 5^2*log(1+x)^2/2! + 31^2*log(1+x)^3/3! + 257^2*log(1+x)^4/4! + ...
where P(x), the partition function of A000041, is generated by:
P(x) = 1 + log(1+x) + 5*log(1+x)^2/2! + 31*log(1+x)^3/3! + 257*log(1+x)^4/4! + ...

Cf. A167137, A000041.

{A167137(n)=sum(k=0,n,numbpart(k)*stirling(n, k, 2)*k!)}
{a(n)=polcoef(sum(m=0,n,A167137(m)^2*log(1+x+x*O(x^n))^m/m!),n)}

a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A167137(k)^2. - Vladeta Jovovic, Nov 08 2009

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)

A306045 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k) / (1 - (exp(x) - 1)^k).

Original entry on oeis.org

1, 2, 10, 74, 682, 7562, 98410, 1463114, 24367402, 449039882, 9069093610, 199050295754, 4713774570922, 119735740542602, 3246094020405610, 93519923311825994, 2852458136048627242, 91805618091515859722, 3108657616523130770410, 110453876295411957125834

Offset: 0

Vaclav Kotesovec, Jun 18 2018

nonn

Convolution of A167137 and A305550.

Vaclav Kotesovec, Table of n, a(n) for n = 0..410

Cf. A015128, A167137, A305550, A306046.

nmax = 20; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k) / (1 - (Exp[x] - 1)^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) = Sum_{k=0..n} Stirling2(n,k) * A015128(k) * k!.

a(n) ~ n! * exp(Pi^2 * (1 - log(2)) / (16*log(2)) + Pi * sqrt(n/(2*log(2)))) / (8*n*(log(2))^n).

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)

A330353 Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).

Original entry on oeis.org

1, 4, 18, 112, 810, 7144, 73458, 850672, 11069370, 161190904, 2575237698, 44571447232, 836188737930, 16970931765064, 368985732635538, 8524290269083792, 208874053200038490, 5428866923032585624, 149250273758730282978, 4318265042184721248352

Offset: 1

Ilya Gutkovskiy, Dec 11 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..420

Cf. A000041, A000203, A000629, A002745, A008277, A038048, A167137, A308555, A330351, A330352, A330354.

nmax = 20; CoefficientList[Series[Sum[(Exp[x] - 1)^k/(k (1 - (Exp[x] - 1)^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] (k - 1)! DivisorSigma[1, k], {k, 1, n}], {n, 1, 20}]

E.g.f.: -Sum_{k>=1} log(1 - (exp(x) - 1)^k).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A167137.
G.f.: Sum_{k>=1} (k - 1)! * sigma(k) * x^k / Product_{j=1..k} (1 - j*x), where sigma = A000203.
exp(Sum_{n>=1} a(n) * log(1 + x)^n / n!) = g.f. of the partition numbers (A000041).
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * sigma(k).
a(n) ~ n! * Pi^2 / (12 * (log(2))^(n+1)). - Vaclav Kotesovec, Dec 14 2019

A305986 Expansion of e.g.f. Product_{k>=1} 1/(1 - (exp(x) - 1)^k/k).

Original entry on oeis.org

1, 1, 4, 21, 144, 1205, 11908, 135597, 1745488, 25045821, 396249564, 6850289765, 128438323720, 2595394603269, 56224162108468, 1299717221807229, 31931915643021504, 830816659779428525, 22820190255069409804, 659845945466402034165, 20034230527927369097848, 637252918691725377815349

Offset: 0

Ilya Gutkovskiy, Jun 15 2018

nonn

Stirling transform of A007841.

Vaclav Kotesovec, Table of n, a(n) for n = 0..420
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A007841, A140585, A167137, A305987.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(combinat[multinomial](n, n-i*j, i$j)*
      b(n-i*j, i-1)*(i-1)!^j, j=0..n/i)))
    end:
a:= n-> add(Stirling2(n, j)*b(j$2), j=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018

nmax = 21; CoefficientList[Series[Product[1/(1 - (Exp[x] - 1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[(Exp[x] - 1)^(j k)/(k j^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
b[0] = 1; b[n_] := b[n] = Sum[(n - 1)!/(n - k)! DivisorSum[k, #^(1 - k/#) &] b[n - k], {k, 1, n}]; a[n_] := a[n] = Sum[StirlingS2[n, k] b[k], {k, 0, n}]; Table[a[n], {n, 0, 21}]

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (exp(x) - 1)^(j*k)/(k*j^k)).
a(n) = Sum_{k=0..n} Stirling2(n,k)*A007841(k).
a(n) ~ c * n! * n / log(2)^n, where c = exp(-gamma) / (4*log(2)^2) = 0.29215... and gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 23 2019

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!.

A306046 Expansion of e.g.f. Product_{k>=1} 1/(1 - (exp(x) - 1)^k)^k.

Original entry on oeis.org

1, 1, 7, 55, 571, 6991, 101467, 1682815, 31370731, 648823951, 14728727227, 363609116575, 9692252794891, 277304683729711, 8471938268282587, 275137855204310335, 9461893931226763051, 343394421233354232271, 13112532730352768439547, 525396814643685317840095

Offset: 0

Vaclav Kotesovec, Jun 18 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..400
Vaclav Kotesovec, Graph - The asymptotic ratio (20000 terms)

Cf. A000219, A167137, A305550, A306045, A306081.

nmax = 20; CoefficientList[Series[Product[1/(1 - (Exp[x] - 1)^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) = Sum_{k=0..n} Stirling2(n,k) * A000219(k) * k!.

a(n) ~ n! * exp(3 * Zeta(3)^(1/3) * n^(2/3) / (2^(4/3) * log(2)^(2/3)) + (1 - log(2)) * Zeta(3)^(2/3) * n^(1/3) / (2^(5/3) * log(2)^(4/3)) - (log(2)^2 + log(2) - 1) * Zeta(3) / (12 * log(2)^2) + 1/12) * Zeta(3)^(7/36) / (A * 2^(11/18) * sqrt(3*Pi) * n^(25/36) * (log(2))^(n + 11/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jun 22 2018

A306022 Stirling transform of partitions numbers (A000041).

Original entry on oeis.org

1, 1, 3, 10, 38, 163, 774, 4006, 22376, 133951, 854402, 5775948, 41190317, 308651432, 2422315371, 19856073597, 169596622997, 1506139073454, 13879704561038, 132488897335228, 1307829322689944, 13330635710335512, 140118664473276174, 1516899115597189064

Offset: 0

Vaclav Kotesovec, Jun 17 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..571
Eric Weisstein's World of Mathematics, Stirling Transform.

Cf. A000041, A167137, A306023.

a:= n-> add(combinat[numbpart](j)*Stirling2(n, j), j=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 17 2018

Table[Sum[StirlingS2[n, k]*PartitionsP[k], {k, 0, n}], {n, 0, 25}]

a(n) = sum(k=0, n, stirling(n, k, 2)*numbpart(k)); \\ Michel Marcus, Jun 17 2018

a(n) = Sum_{k=0..n} Stirling2(n,k)*A000041(k).

A316143 Expansion of e.g.f. Product_{k>=1} 1 / (1 - (exp(x)-1)^k)^2.

Original entry on oeis.org

1, 2, 12, 92, 912, 10772, 148512, 2328692, 40842912, 791302772, 16767551712, 385382491892, 9542377300512, 253105962752372, 7156766466076512, 214814484529608692, 6819311473596695712, 228212485803422931572, 8028037725386962194912, 296094910181041530831092

Offset: 0

Vaclav Kotesovec, Jun 25 2018

Self-convolution of A167137.

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 7 we obtain the sequence [1, 2, 5, 1, 2, 6, 0, 2, 5, 1, 2, 6, 0, 2, 5, 1, 2, 6, 0, ...], with a preperiod of length 1 and an apparent period thereafter of 6 = phi(7). - Peter Bala, Mar 03 2023

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

Cf. A000712, A167137, A316142, A316144.

nmax = 20; CoefficientList[Series[Product[1/(1-(Exp[x]-1)^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

Sum_{k=0..n} binomial(n,k) * A167137(k) * A167137(n-k).

a(n) ~ n! * exp(Pi * sqrt(2*n/(3*log(2))) - Pi^2 * (1 - 1/log(2)) / 12) / (2^(7/4) * 3^(3/4) * n^(5/4) * (log(2))^(n - 1/4)).

cp's OEIS Frontend

A167138 G.f.: Sum_{n>=0} A167137(n)^2 * log(1+x)^n/n! where Sum_{n>=0} A167137(n)*log(1+x)^n/n! = g.f. of the partition numbers (A000041).

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

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A306045 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k) / (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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A330353 Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).

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A305986 Expansion of e.g.f. Product_{k>=1} 1/(1 - (exp(x) - 1)^k/k).

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

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A306046 Expansion of e.g.f. Product_{k>=1} 1/(1 - (exp(x) - 1)^k)^k.

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