A007840 -id:A007840 - cp's OEIS frontend

A367139 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x)^3)).

1, 1, 9, 167, 4780, 186004, 9173780, 548563140, 38573633016, 3119384230176, 285237426927552, 29102185296785160, 3277703460197645232, 403931173342682581296, 54066960915411480743520, 7811249803193620134996864, 1211525560869437165319590400

Offset: 0

Seiichi Manyama, Nov 06 2023

nonn

Cf. A007840, A052802, A367138.

Cf. A367135, A367137.

Table[1/(3*n+1)! * Sum[(3*n+k)! * Abs[StirlingS1[n,k]], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)

a(n) = sum(k=0, n, (3*n+k)!*abs(stirling(n, k, 1)))/(3*n+1)!;

a(n) = (1/(3*n+1)!) * Sum_{k=0..n} (3*n+k)! * |Stirling1(n,k)|.

a(n) ~ LambertW(3*exp(4))^n * n^(n-1) / (sqrt(3*(1 + LambertW(3*exp(4)))) * exp(n) * (-3 + LambertW(3*exp(4)))^(4*n + 1)). - Vaclav Kotesovec, Nov 07 2023

A305306 Expansion of e.g.f. 1/(1 + log(1 - x)/(1 - x)).

Original entry on oeis.org

1, 1, 5, 35, 324, 3744, 51902, 839362, 15513096, 322550616, 7451677632, 189366303840, 5249764639248, 157666361452560, 5099445234111888, 176713626295062384, 6531995374500741888, 256537368987293878272, 10667901271715707803264, 468261481657502075856768, 21635865693957558515860224

Offset: 0

Ilya Gutkovskiy, May 29 2018

nonn

a(n)/n! is the invert transform of [1, 1 + 1/2, 1 + 1/2 + 1/3, 1 + 1/2 + 1/3 + 1/4, ...].

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 35*x^3/3! + 324*x^4/4! + 3744*x^5/5! + 51902*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..395
N. J. A. Sloane, Transforms

Cf. A000254, A001008, A002805, A007840, A128044, A128045, A305307.

Cf. A006153, A087761.

H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
b:= proc(n) option remember; `if`(n=0, 1,
      add(H(j)*b(n-j), j=1..n))
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..20);  # Alois P. Heinz, May 29 2018

nmax = 20; CoefficientList[Series[1/(1 + Log[1 - x]/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[1/(1 - Sum[HarmonicNumber[k] x^k , {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[HarmonicNumber[k] a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 20}]

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)/(1-x)))) \\ Seiichi Manyama, May 10 2023

E.g.f.: 1/(1 - Sum_{k>=1} (A001008(k)/A002805(k))*x^k).
a(n) ~ n! / ((1/LambertW(1)^2 - 1) * (1 - LambertW(1))^n). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A006153(k). - Seiichi Manyama, May 10 2023

A367138 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x)^2)).

Original entry on oeis.org

1, 1, 7, 98, 2096, 60684, 2221766, 98488592, 5129567208, 307066395000, 20775900638472, 1567955813868960, 130596146677118448, 11899839375083061024, 1177540373453616858240, 125754589311488009416704, 14416305655742615673941760, 1765794816084642802179333120

Offset: 0

Seiichi Manyama, Nov 06 2023

nonn

Cf. A007840, A052802, A367139.

Cf. A367134, A367136.

Table[1/(2*n+1)! * Sum[(2*n+k)! * Abs[StirlingS1[n,k]], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)

a(n) = sum(k=0, n, (2*n+k)!*abs(stirling(n, k, 1)))/(2*n+1)!;

a(n) = (1/(2*n+1)!) * Sum_{k=0..n} (2*n+k)! * |Stirling1(n,k)|.

a(n) ~ LambertW(2*exp(3))^n * n^(n-1) / (sqrt(2*(1 + LambertW(2*exp(3)))) * exp(n) * (-2 + LambertW(2*exp(3)))^(3*n + 1)). - Vaclav Kotesovec, Nov 07 2023

A330149 Expansion of e.g.f. exp(-x) / (1 + log(1 - x)).

Original entry on oeis.org

1, 0, 2, 7, 47, 368, 3494, 38673, 489341, 6966344, 110199090, 1917589771, 36402276107, 748629861016, 16580304397942, 393443385034069, 9958671117295737, 267824225078212336, 7626444798009902530, 229232204568273395919, 7252798333599466521575, 240948882537990850397536

Offset: 0

Ilya Gutkovskiy, Dec 03 2019

nonn

Inverse binomial transform of A007840.

Cf. A052841, A330150.

Cf. A007840, A291979.

nmax = 21; CoefficientList[Series[Exp[-x]/(1 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k) * A007840(k).
a(n) ~ n! * exp(n + exp(-1) - 1) / (exp(1) - 1)^(n+1). - Vaclav Kotesovec, Dec 15 2019
a(n) = (-1)^n + Sum_{k=1..n} (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Dec 19 2023

A346987 Expansion of e.g.f. 1 / (1 + 5 * log(1 - x))^(1/5).

Original entry on oeis.org

1, 1, 7, 86, 1524, 35370, 1015590, 34757400, 1381147440, 62498177880, 3172764322680, 178566159846480, 11034757650750960, 742773843654742080, 54094804600076176320, 4238009228531321452800, 355400361455423327193600, 31764402860426288679456000, 3014207878695233997923193600

Offset: 0

Ilya Gutkovskiy, Aug 11 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..350

Cf. A007840, A008548, A346978, A346984, A347015, A347016, A347019.

nmax = 18; CoefficientList[Series[1/(1 + 5 Log[1 - x])^(1/5), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] 5^k Pochhammer[1/5, k], {k, 0, n}], {n, 0, 18}]

a[n]:=if n=0 then 1 else sum(n!/(n-k)!*(5/k-4/n)*a[n-k],k,1,n);
makelist(a[n],n,0,50); /* Tani Akinari, Aug 27 2023 */

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A008548(k).
a(n) ~ n! * exp(n/5) / (Gamma(1/5) * 5^(1/5) * n^(4/5) * (exp(1/5) - 1)^(n + 1/5)). - Vaclav Kotesovec, Aug 14 2021
For n > 0, a(n) = Sum_{k=1..n} (n!/(n-k)!)*(5/k-4/n)*a(n-k). - Tani Akinari, Aug 27 2023

A347015 Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^(1/3).

Original entry on oeis.org

1, 1, 5, 42, 498, 7644, 144156, 3225648, 83536008, 2457701928, 80970232104, 2953056534768, 118112744060208, 5140622709134496, 241863782829704928, 12232551538417012992, 661818290353375962240, 38140594162828447248000, 2332567001993176540206720, 150880256846462633823648000

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..372

Cf. A007559, A007840, A346978, A346982, A347016.

Cf. A354263.

g:= proc(n) option remember; `if`(n<2, 1, (3*n-2)*g(n-1)) end:
a:= n-> add(abs(Stirling1(n, k))*g(k), k=0..n):
seq(a(n), n=0..19);  # Alois P. Heinz, Aug 10 2021

nmax = 19; CoefficientList[Series[1/(1 + 3 Log[1 - x])^(1/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 19}]

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A007559(k).
a(n) ~ n! * exp(n/3) / (Gamma(1/3) * 3^(1/3) * n^(2/3) * (exp(1/3) - 1)^(n + 1/3)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (3 - 2*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023

A336259 a(0) = 1; a(n) = (n!)^3 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^3.

Original entry on oeis.org

1, 1, 9, 278, 20464, 2948824, 735078968, 291153023664, 172201253334528, 145044581320046592, 167609226267379703808, 257816558769660828601344, 514890814087717253133447168, 1307445058678686737908660752384, 4146656933568759002389401276616704

Offset: 0

Ilya Gutkovskiy, Jul 15 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..180

Cf. A007840, A193436, A336195, A336258, A336260, A336261.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)/i^3, i=1..n))
    end:
a:= n-> n!^3*b(n):
seq(a(n), n=0..14);  # Alois P. Heinz, Jan 04 2024

a[0] = 1; a[n_] := a[n] = (n!)^3 Sum[a[k]/(k! (n - k))^3, {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]
nmax = 14; CoefficientList[Series[1/(1 - PolyLog[3, x]), {x, 0, nmax}], x] Range[0, nmax]!^3

a(n) = (n!)^3 * [x^n] 1 / (1 - polylog(3,x)).

a(n) ~ (n!)^3 / (polylog(2,r) * r^n), where r = 0.86512013798076629268795131756... is the root of the equation polylog(3,r) = 1. - Vaclav Kotesovec, Jul 15 2020

A336260 a(0) = 1; a(n) = (n!)^4 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^4.

Original entry on oeis.org

1, 1, 17, 1474, 404768, 271581776, 377987513392, 974814164752800, 4289222350867156992, 30232332223815625555968, 324796212685273837095714816, 5108947647642107040382284423168, 113818571142935411070742114448769024, 3492592855002964381945529723625305210880

Offset: 0

Ilya Gutkovskiy, Jul 15 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..143

Cf. A007840, A217145, A336196, A336258, A336259, A336261.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)/i^4, i=1..n))
    end:
a:= n-> n!^4*b(n):
seq(a(n), n=0..14);  # Alois P. Heinz, Jan 04 2024

a[0] = 1; a[n_] := a[n] = (n!)^4 Sum[a[k]/(k! (n - k))^4, {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]
nmax = 13; CoefficientList[Series[1/(1 - PolyLog[4, x]), {x, 0, nmax}], x] Range[0, nmax]!^4

a(n) = (n!)^4 * [x^n] 1 / (1 - polylog(4,x)).

a(n) ~ (n!)^4 / (polylog(3,r) * r^n), where r = 0.93073451517099234709643607941... is the root of the equation polylog(4,r) = 1. - Vaclav Kotesovec, Jul 15 2020

A336261 a(0) = 1; a(n) = (n!)^5 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^5.

Original entry on oeis.org

1, 1, 33, 8294, 8790208, 28436662624, 228929520628448, 3983602580423420352, 135150778123405293748224, 8262821715336263175482769408, 855516444430388524429593124012032, 142657102263368111456587968163250896896, 36753801552552818015956675623665562408714240

Offset: 0

Ilya Gutkovskiy, Jul 15 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..120

Cf. A007840, A336197, A336258, A336259, A336260.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)/i^5, i=1..n))
    end:
a:= n-> n!^5*b(n):
seq(a(n), n=0..14);  # Alois P. Heinz, Jan 04 2024

a[0] = 1; a[n_] := a[n] = (n!)^5 Sum[a[k]/(k! (n - k))^5, {k, 0, n - 1}]; Table[a[n], {n, 0, 12}]
nmax = 12; CoefficientList[Series[1/(1 - PolyLog[5, x]), {x, 0, nmax}], x] Range[0, nmax]!^5

a(n) = (n!)^5 * [x^n] 1 / (1 - polylog(5,x)).

a(n) ~ (n!)^5 / (polylog(4,r) * r^n), where r = 0.96581751668950729310276791428... is the root of the equation polylog(5,r) = 1. - Vaclav Kotesovec, Jul 15 2020

A347016 Expansion of e.g.f. 1 / (1 + 4 * log(1 - x))^(1/4).

Original entry on oeis.org

1, 1, 6, 62, 916, 17644, 419360, 11859840, 388965600, 14514046560, 607165485120, 28143329181120, 1431690475207680, 79302863940387840, 4751108622148907520, 306118435580577146880, 21107196651940518551040, 1550773243761690603179520, 120947288498720390755353600

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..360

Cf. A007696, A007840, A346978, A346983, A347015.

g:= proc(n) option remember; `if`(n<2, 1, (4*n-3)*g(n-1)) end:
a:= n-> add(abs(Stirling1(n, k))*g(k), k=0..n):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 10 2021

nmax = 18; CoefficientList[Series[1/(1 + 4 Log[1 - x])^(1/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] 4^k Pochhammer[1/4, k], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A007696(k).
a(n) ~ n! * exp(n/4) / (Gamma(1/4) * 2^(1/2) * n^(3/4) * (exp(1/4) - 1)^(n + 1/4)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (4 - 3*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023

cp's OEIS Frontend

A367139 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x)^3)).

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A305306 Expansion of e.g.f. 1/(1 + log(1 - x)/(1 - x)).

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A367138 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x)^2)).

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A330149 Expansion of e.g.f. exp(-x) / (1 + log(1 - x)).

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A346987 Expansion of e.g.f. 1 / (1 + 5 * log(1 - x))^(1/5).

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A347015 Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^(1/3).

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A336259 a(0) = 1; a(n) = (n!)^3 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^3.

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A336260 a(0) = 1; a(n) = (n!)^4 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^4.

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