A007840 -id:A007840 - cp's OEIS frontend

A346921 Expansion of e.g.f. 1 / (1 - log(1 - x)^2 / 2).

1, 0, 1, 3, 17, 110, 874, 8064, 85182, 1012248, 13369026, 194245590, 3079135806, 52880064588, 978038495316, 19381794788160, 409702099828104, 9201877089355584, 218832476773294008, 5493266481129425064, 145153549897858762776, 4027310838211114515600

Offset: 0

Ilya Gutkovskiy, Aug 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..430

Cf. A007840, A346922, A346923, A346924.

Cf. A000254, A052811, A305306, A330047, A347001.

nmax = 21; CoefficientList[Series[1/(1 - Log[1 - x]^2/2), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 2]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1-log(1-x)^2/2))) \\ Michel Marcus, Aug 07 2021

a(n) = sum(k=0, n\2, (2*k)!*abs(stirling(n, 2*k, 1))/2^k); \\ Seiichi Manyama, May 06 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,2)| * a(n-k).
a(n) ~ n! * exp(sqrt(2)*n) / (sqrt(2) * (exp(sqrt(2)) - 1)^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * |Stirling1(n,2*k)|/2^k. - Seiichi Manyama, May 06 2022

A346922 Expansion of e.g.f. 1 / (1 + log(1 - x)^3 / 3!).

Original entry on oeis.org

1, 0, 0, 1, 6, 35, 245, 2044, 19572, 210524, 2513760, 33012276, 472963876, 7340889192, 122703087416, 2197496734224, 41979155247520, 852063971170960, 18312093589455440, 415420659953439840, 9920128280950954080, 248735658391768241280, 6533773435848445617600

Offset: 0

Ilya Gutkovskiy, Aug 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..437

Cf. A007840, A346921, A346923, A346924.

Cf. A000399, A346894, A347002, A353118.

nmax = 22; CoefficientList[Series[1/(1 + Log[1 - x]^3/3!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 3]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-x)^3/3!))) \\ Michel Marcus, Aug 07 2021

a(n) = sum(k=0, n\3, (3*k)!*abs(stirling(n, 3*k, 1))/6^k); \\ Seiichi Manyama, May 06 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,3)| * a(n-k).
a(n) ~ n! * 6^(1/3) / (3 * exp(6^(1/3)) * (1 - exp(-6^(1/3)))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * |Stirling1(n,3*k)|/6^k. - Seiichi Manyama, May 06 2022

A354122 Expansion of e.g.f. 1/(1 + log(1 - x))^3.

Original entry on oeis.org

1, 3, 15, 102, 870, 8892, 105708, 1431168, 21722136, 365105928, 6729341832, 134915992560, 2922576142320, 68013701197920, 1692075061072800, 44810389419079680, 1258472984174461440, 37357062009383877120, 1168635883239630120960, 38424619272539153157120

Offset: 0

Seiichi Manyama, May 17 2022

nonn

Cf. A007840, A052801, A354123.

Cf. A226515, A354120.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x))^3))

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

a(n) = (1/2) * Sum_{k=0..n} (k + 2)! * |Stirling1(n,k)|.
a(n) ~ sqrt(Pi/2) * n^(n + 5/2) / (exp(1) - 1)^(n+3). - Vaclav Kotesovec, Jun 04 2022
a(0) = 1; a(n) = Sum_{k=1..n} (2*k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A382792 a(n) = Sum_{k=0..n} (Stirling1(n,k) * k!)^2.

Original entry on oeis.org

1, 1, 5, 76, 2392, 126676, 10057204, 1114096320, 163918005696, 30894047577216, 7254176241285504, 2075722128162164736, 710883208780304954112, 287061726161439955116288, 134961239570613490548986112, 73079781978184515947237031936, 45150931601954398539342470578176

Offset: 0

Ilya Gutkovskiy, Apr 05 2025

nonn

In general, for m>=1, Sum_{k=0..n} (abs(Stirling1(n,k)) * k!)^m ~ sqrt(2*Pi/m) * n^(m*n + 1/2) / (exp(1) - 1)^(m*n+1). - Vaclav Kotesovec, Apr 05 2025

Main diagonal of A379821.

Cf. A006252, A007840, A047796, A048144, A382794.

Table[Sum[(StirlingS1[n, k] k!)^2, {k, 0, n}], {n, 0, 16}]
Table[(n!)^2 SeriesCoefficient[1/(1 - Log[1 + x] Log[1 + y]), {x, 0, n}, {y, 0, n}], {n, 0, 16}]

a(n) = sum(k=0, n, (k!*stirling(n, k, 1))^2); \\ Seiichi Manyama, Apr 05 2025

a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1 + x) * log(1 + y)).
a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1 - x) * log(1 - y)).
a(n) ~ sqrt(Pi) * n^(2*n + 1/2) / (exp(1) - 1)^(2*n+1). - Vaclav Kotesovec, Apr 05 2025

A226226 Number of alignments of n points with no singleton cycles.

Original entry on oeis.org

1, 0, 1, 2, 12, 64, 470, 3828, 36456, 387840, 4603392, 60061440, 855664656, 13207470912, 219609303888, 3912940891104, 74377769483520, 1502277409668096, 32130095812624512, 725400731911792896, 17240044059713320704, 430231117562438446080, 11248105572520779755520

Offset: 0

Ricardo Bittencourt, May 31 2013

nonn

a(n) is the number of labeled sequences of cycles, where no cycle has size 1.

			For n=4, the a(4)=12 alignments with no singletons are: 1234, 1243, 1324, 1342, 1423, 1432, 12|34, 13|24, 14|23, 23|14, 24|13, 34|12.

P. Flajolet and R. Segdewick, Analytic Combinatorics, Cambridge University Press, 2009, page 119

Vincenzo Librandi, Table of n, a(n) for n = 0..200

The alignments with singletons included are given by A007840.

Range[0, 50]! CoefficientList[ Series[(1 + z - Log[1/(1 - z)])^(-1), {z, 0, 50}], z]

x='x+O('x^66); Vec(serlaplace(1/(1+x-log(1/(1-x))))) \\ Joerg Arndt, Jun 01 2013

E.g.f.: 1/(1+x-log(1/(1-x)))
a(n) ~ n!*c/(1-c)^(n+2), where c = -LambertW(-exp(-2)) = 0.158594339563... - Vaclav Kotesovec, Jun 02 2013
a(0) = 1; a(n) = Sum_{k=0..n-2} binomial(n,k) * (n-k-1)! * a(k). - Ilya Gutkovskiy, Apr 26 2021

A323339 Numerator of the sum of inverse products of parts in all compositions of n.

Original entry on oeis.org

1, 1, 3, 7, 11, 347, 3289, 1011, 38371, 136553, 4320019, 12528587, 40771123, 29346499543, 129990006917, 1927874590951, 903657004321, 437445829053473, 12456509813711881, 187206004658210129, 1974369484466728177, 1967745662306280217, 21401375717067880189

Offset: 0

Alois P. Heinz, Jan 11 2019

Numerators of the INVERT transform of reciprocal integers.

			1/1, 1/1, 3/2, 7/3, 11/3, 347/60, 3289/360, 1011/70, 38371/1680, 136553/3780, 4320019/75600, 12528587/138600, 40771123/285120, ... = A323339/A323340

Alois P. Heinz, Table of n, a(n) for n = 0..466

Denominators: A323340.

Cf. A000142, A007840, A011782, A088305, A177208, A177209, A322364, A322365, A322380, A322381, A323290, A323291.

b:= proc(n) option remember;
     `if`(n=0, 1, add(b(n-j)/j, j=1..n))
    end:
a:= n-> numer(b(n)):
seq(a(n), n=0..25);

nmax = 20; Numerator[CoefficientList[Series[1/(1 + Log[1-x]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Feb 12 2024 *)

G.f. for fractions: 1 / (1 + log(1 - x)). - Ilya Gutkovskiy, Nov 12 2019
a(n) = numerator( A007840(n)/n! ). - Alois P. Heinz, Jan 04 2024
A323339(n)/A323340(n) ~ exp(n) / (exp(1) - 1)^(n+1). - Vaclav Kotesovec, Feb 12 2024

A323340 Denominator of the sum of inverse products of parts in all compositions of n.

Original entry on oeis.org

1, 1, 2, 3, 3, 60, 360, 70, 1680, 3780, 75600, 138600, 285120, 129729600, 363242880, 3405402000, 1009008000, 308756448000, 5557616064000, 52797352608000, 351982350720000, 221748880953600, 1524523556556000, 738190353700800, 13464592051502592000

Offset: 0

Alois P. Heinz, Jan 11 2019

Denominators of the INVERT transform of reciprocal integers.

Alois P. Heinz, Table of n, a(n) for n = 0..506

See A323339 for more information.

Cf. A000142, A007840.

b:= proc(n) option remember;
     `if`(n=0, 1, add(b(n-j)/j, j=1..n))
    end:
a:= n-> denom(b(n)):
seq(a(n), n=0..25);

nmax = 20; Denominator[CoefficientList[Series[1/(1 + Log[1-x]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Feb 12 2024 *)

a(n) = denominator( A007840(n)/n! ).

A346923 Expansion of e.g.f. 1 / (1 - log(1 - x)^4 / 4!).

Original entry on oeis.org

1, 0, 0, 0, 1, 10, 85, 735, 6839, 69804, 784580, 9680000, 130312336, 1901581968, 29895585356, 503657235900, 9051009737834, 172807817059664, 3493189152511608, 74530548004474584, 1673793045085649146, 39467836062718058100, 974939402596817961050, 25177327470510057799550

Offset: 0

Ilya Gutkovskiy, Aug 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..441

Cf. A007840, A346921, A346922, A346924.

Cf. A000454, A346895, A347003, A353119.

nmax = 23; CoefficientList[Series[1/(1 - Log[1 - x]^4/4!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 4]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1-log(1-x)^4/4!))) \\ Michel Marcus, Aug 07 2021

a(n) = sum(k=0, n\4, (4*k)!*abs(stirling(n, 4*k, 1))/24^k); \\ Seiichi Manyama, May 06 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,4)| * a(n-k).
a(n) ~ n! * 2^(-5/4) * 3^(1/4) / (exp(2^(3/4)*3^(1/4)) * (1 - exp(-2^(3/4)*3^(1/4)))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * |Stirling1(n,4*k)|/24^k. - Seiichi Manyama, May 06 2022

A346924 Expansion of e.g.f. 1 / (1 + log(1 - x)^5 / 5!).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 15, 175, 1960, 22449, 269577, 3430790, 46480830, 671260876, 10329270952, 169125055736, 2940784282800, 54182845939104, 1055291277366108, 21674715826211532, 468366193441002564, 10624074081842024496, 252432685158931968768, 6270222495850552958004

Offset: 0

Ilya Gutkovskiy, Aug 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..444

Cf. A007840, A346921, A346922, A346923.

Cf. A000482, A347004, A353200.

nmax = 23; CoefficientList[Series[1/(1 + Log[1 - x]^5/5!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 5]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-x)^5/5!))) \\ Michel Marcus, Aug 07 2021

a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1))/120^k); \\ Seiichi Manyama, May 06 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,5)| * a(n-k).
a(n) ~ n! * 2^(3/5) * 3^(1/5) * exp(2^(3/5)*15^(1/5)*n) / (5^(4/5) * (exp(2^(3/5)*15^(1/5)) - 1)^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*k)|/120^k. - Seiichi Manyama, May 06 2022

A346978 Expansion of e.g.f. 1 / sqrt(1 + 2 * log(1 - x)).

Original entry on oeis.org

1, 1, 4, 26, 234, 2694, 37812, 626352, 11962164, 258787812, 6255195168, 167072685240, 4886611129320, 155335056242040, 5332298685827760, 196590247328769120, 7747254471910795920, 324986515253994589200, 14458392906960271354560, 679977065168639138610720

Offset: 0

Ilya Gutkovskiy, Aug 09 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..390

Cf. A001147, A007840, A088500, A305404, A320343.

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

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

cp's OEIS Frontend

A346921 Expansion of e.g.f. 1 / (1 - log(1 - x)^2 / 2).

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A346922 Expansion of e.g.f. 1 / (1 + log(1 - x)^3 / 3!).

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A354122 Expansion of e.g.f. 1/(1 + log(1 - x))^3.

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A382792 a(n) = Sum_{k=0..n} (Stirling1(n,k) * k!)^2.

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A226226 Number of alignments of n points with no singleton cycles.

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A323339 Numerator of the sum of inverse products of parts in all compositions of n.

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A323340 Denominator of the sum of inverse products of parts in all compositions of n.

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A346923 Expansion of e.g.f. 1 / (1 - log(1 - x)^4 / 4!).

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