A305550 -id:A305550 - 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)

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).

A305987 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k/k).

Original entry on oeis.org

1, 1, 2, 9, 52, 355, 2976, 29897, 343988, 4423503, 63088600, 992691205, 17095554444, 319404545291, 6427307831504, 138546745515393, 3185841858310180, 77866726065935239, 2016161715005701128, 55127056896177521981, 1587073087715010466556, 47982707153606476112067, 1519931218769637781731712

Offset: 0

Ilya Gutkovskiy, Jun 15 2018

nonn

Stirling transform of A007838.

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. A007838, A305547, A305550, A305986.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(combinat[multinomial](n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*(i-1)!^j, j=0..min(1, 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 = 22; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) (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, 22}]

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*(exp(x) - 1)^(j*k)/(k*j^k)).
a(n) = Sum_{k=0..n} Stirling2(n,k)*A007838(k).
a(n) ~ exp(-gamma) * n! / (2 * log(2)^(n+1)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 23 2019

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

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)

A306023 Stirling transform of partitions into distinct parts (A000009).

Original entry on oeis.org

1, 1, 2, 6, 22, 89, 391, 1875, 9822, 55817, 340535, 2208681, 15118109, 108677575, 817914056, 6431115486, 52741729600, 450432487463, 3999401133601, 36853795902353, 351799243932131, 3472526583025397, 35382850151528847, 371592232539942447, 4016792440158613798

Offset: 0

Vaclav Kotesovec, Jun 17 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..573
Eric Weisstein's World of Mathematics, Stirling Transform.

Cf. A000009, A305550, A306022.

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(b(j)*Stirling2(n, j), j=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 17 2018

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

a(n) = Sum_{k=0..n} Stirling2(n,k)*A000009(k).

A306081 Expansion of e.g.f. Product_{k>=1} ((1 + (exp(x) - 1)^k) / (1 - (exp(x) - 1)^k))^k.

Original entry on oeis.org

1, 2, 14, 134, 1574, 22262, 367814, 6907574, 144942854, 3357588662, 85000841414, 2331998188214, 68862337593734, 2176283210561462, 73250933670041414, 2614843434740912054, 98632371931151518214, 3918608865052986708662, 163507638190268814991814

Offset: 0

Vaclav Kotesovec, Jun 20 2018

nonn

Convolution of A306080 and A306046.

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

Cf. A156616, A305550, A306045, A306046, A306080.

nmax = 20; CoefficientList[Series[Product[((1 + (Exp[x] - 1)^k)/(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) * A156616(k) * k!.

a(n) ~ n! * exp(3 * (7*Zeta(3))^(1/3) * n^(2/3) / (4 * log(2)^(2/3)) + (1 - log(2)) * (7*Zeta(3))^(2/3) * n^(1/3) / (8 * log(2)^(4/3)) - 7*(log(2)^2 + log(2) - 1) * Zeta(3) / (48 * log(2)^2) + 1/12) * (7*Zeta(3))^(7/36) / (A * 2^(13/12) * sqrt(3*Pi) * n^(25/36) * (log(2))^(n + 11/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jun 22 2018

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!.

A316142 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x)-1)^k)^2.

Original entry on oeis.org

1, 2, 8, 56, 476, 4832, 58508, 815936, 12750956, 220610432, 4195325708, 86976996416, 1949966347436, 46965887762432, 1208922621624908, 33111231803362496, 961354836530983916, 29490401681798152832, 952900154176192244108, 32342850619899263226176

Offset: 0

Vaclav Kotesovec, Jun 25 2018

Self-convolution of A305550.

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, 1, 0, 0, 2, 2, 2, 1, 0, 0, 2, 2, 2, 1, 0, 0, 2, 2, ...], 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. A022567, A305550, A316143, A316144.

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

Sum_{k=0..n} binomial(n,k) * A305550(k) * A305550(n-k).

a(n) ~ n! * exp(Pi * sqrt(n/(3*log(2))) - Pi^2 * (1 - 1/log(2)) / 24) / (2^(5/2) * 3^(1/4) * (log(2))^(n + 1/4) * n^(3/4)).

A326884 E.g.f.: Product_{k>=1} (1 + k*(exp(x)-1)^k).

Original entry on oeis.org

1, 1, 5, 43, 377, 4291, 58745, 914803, 15641897, 298104451, 6337624985, 147137420563, 3674045105417, 98093008751011, 2793940490888825, 84812168406518323, 2737609202984488937, 93486719521251467971, 3358396276982001106265, 126434158646122122080083

Offset: 0

Vaclav Kotesovec, Jul 31 2019

nonn

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

Cf. A022629, A305550, A305987, A326885.

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

a(n) = Sum_{k=0..n} A022629(k)*Stirling2(n,k)*k!.

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

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A305987 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k/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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A320350 Expansion of e.g.f. Product_{k>=1} (1 + log(1/(1 - x))^k).

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A306023 Stirling transform of partitions into distinct parts (A000009).

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A306081 Expansion of e.g.f. Product_{k>=1} ((1 + (exp(x) - 1)^k) / (1 - (exp(x) - 1)^k))^k.

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

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