A306045 -id:A306045 - 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)

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)

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

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

A316144 Expansion of e.g.f. Product_{k>=1} ((1 + (exp(x)-1)^k) / (1 - (exp(x)-1)^k))^2.

Original entry on oeis.org

1, 4, 28, 268, 3148, 43564, 692428, 12390508, 245896588, 5351817004, 126614238028, 3232332423148, 88500275727628, 2585371577628844, 80227707005300428, 2634361286274638188, 91223969834203056268, 3321457538305952791084, 126817592900018186967628

Offset: 0

Vaclav Kotesovec, Jun 25 2018

Self-convolution of A306045.
Convolution of A316142 and A316143.
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, 4, 0, 2, 5, 3, 2, 4, 0, 2, 5, 3, 2, 4, 0, 2, 5, 3, 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. A001934, A306045, A316142, A316143.

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

Sum_{k=0..n} binomial(n,k) * A306045(k) * A306045(n-k).

a(n) ~ n! * exp(Pi * sqrt(n/log(2)) - Pi^2 * (1 - 1/log(2)) / 8) / (2^(7/2) * n^(5/4) * log(2)^(n - 1/4)).

A307523 Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k)/(1 - log(1 + x)^k).

Original entry on oeis.org

1, 2, 6, 28, 124, 848, 5312, 40080, 367632, 3132096, 27731328, 474979008, 1130161728, 90279554688, 268809015168, 3005011325952, 473192066191104, -7913323872693504, 186235895195313408, 1357401816746159616, -181477915903332002304, 9552839425392612096000

Offset: 0

Ilya Gutkovskiy, Apr 12 2019

sign

Exponential convolution of A298905 and A306042.

Cf. A015128, A298905, A306042, A306045, A307524.

nmax = 21; CoefficientList[Series[Product[(1 + Log[1 + x]^k)/(1 - Log[1 + x]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2 k] - DivisorSigma[1, k]) Log[1 + x]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/EllipticTheta[4, 0, Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] Sum[PartitionsQ[j] PartitionsP[k - j], {j, 0, k}] k!, {k, 0, n}], {n, 0, 21}]

E.g.f.: exp(Sum_{k>=1} (sigma(2*k) - sigma(k))*log(1 + x)^k/k).
E.g.f.: 1/theta_4(log(1 + x)).
a(n) = Sum_{k=0..n} Stirling1(n,k)*A015128(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

A326886 E.g.f.: Product_{k>=1} (1 + k(exp(x)-1)^k) / (1 - k(exp(x)-1)^k).

Original entry on oeis.org

1, 2, 14, 134, 1574, 22262, 370694, 7008374, 147805574, 3447703862, 88047037574, 2438080410614, 72703788119174, 2321967591003062, 79030014919422854, 2854499200663284854, 109018338380110506374, 4388176453133542327862, 185612789014681549094534

Offset: 0

Vaclav Kotesovec, Jul 31 2019

nonn

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

Cf. A265758, A306045, A326884, A326885, A326887.

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

a(n) = Sum_{k=0..n} A265758(k)*Stirling2(n,k)*k!.

a(n) ~ c * 2 * (3^(2/3) + 2) * n! / (3*(3^(2/3) - 2) * (3^(1/3) - 1) * log(1 + 3^(-1/3))^(n+1)), where c = Product_{k>=4} (1 + k/3^(k/3)) / (1 - k/3^(k/3)) = 153073.83255100475812062139772279157814388739...

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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A305550 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^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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A306081 Expansion of e.g.f. Product_{k>=1} ((1 + (exp(x) - 1)^k) / (1 - (exp(x) - 1)^k))^k.

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A316144 Expansion of e.g.f. Product_{k>=1} ((1 + (exp(x)-1)^k) / (1 - (exp(x)-1)^k))^2.

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A307523 Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k)/(1 - log(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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A326886 E.g.f.: Product_{k>=1} (1 + k*(exp(x)-1)^k) / (1 - k*(exp(x)-1)^k).

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A326886 E.g.f.: Product_{k>=1} (1 + k(exp(x)-1)^k) / (1 - k(exp(x)-1)^k).