A140585 -id:A140585 - cp's OEIS frontend

A144792 EXP transform of A140585.

1, 1, 5, 33, 282, 2938, 36029, 507440, 8058990, 142315830, 2763775025, 58498072273, 1339545500214, 32980132065364, 868417100538399, 24344702489881998, 723694354351500431, 22733368105181643193, 752291980101845144878, 26153153055424960528533

Offset: 0

Thomas Wieder, Sep 21 2008

nonn

Stirling transform of A143463.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A140585, A143463.

with(numtheory): with(combinat): b:= proc(k) option remember; add(d/d!^(k/d), d=divisors(k)) end: c:= proc(n) option remember; `if`(n=0, 1, add((n-1)!/(n-k)!* b(k)* c(n-k), k=1..n)) end: aa:= n-> add(stirling2(n, k) *c(k), k=1..n): a:= proc(n) option remember; `if`(n=0, 1, aa(n)+ add(binomial(n-1, k-1) *aa(k) *a(n-k), k=1..n-1)) end: seq(a(n), n=1..20); # Alois P. Heinz, Oct 10 2008

b[k_] := b[k] = DivisorSum[k, #/#!^(k/#)&]; c[n_] := c[n] = If[n==0, 1, Sum[(n-1)!/(n-k)!*b[k]*c[n-k], {k, 1, n}]]; aa[n_] := Sum[StirlingS2[n, k]*c[k], {k, 1, n}]; a[n_] := a[n] = If[n==0, 1, aa[n] + Sum[Binomial[ n-1, k-1]*aa[k]*a[n-k], {k, 1, n-1}]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 25 2017, after Alois P. Heinz *)

E.g.f: (1/exp(1)) exp( 1 / prod_{k=1}^{inf} (1 - (exp(x)-1)^k / k!) ).
a(n) = sum_{k=1..n} C(n-1,k-1) A140585(k) a(n-k).
With S2(n,k) as the Stirling number of the second kind we have
a(n) = sum_{k=1..n} A143463(n) S2(n,k).

More terms from Alois P. Heinz, Oct 10 2008

A005651 Sum of multinomial coefficients (n_1+n_2+...)!/(n_1!n_2!...) where (n_1, n_2, ...) runs over all integer partitions of n.

Original entry on oeis.org

1, 1, 3, 10, 47, 246, 1602, 11481, 95503, 871030, 8879558, 98329551, 1191578522, 15543026747, 218668538441, 3285749117475, 52700813279423, 896697825211142, 16160442591627990, 307183340680888755, 6147451460222703502, 129125045333789172825, 2841626597871149750951

Offset: 0

Simon Plouffe

This is the total number of hierarchies of n labeled elements arranged on 1 to n levels. A distribution of elements onto levels is "hierarchical" if a level l+1 contains <= elements than level l. Thus for n=4 the arrangement {1,2}:{3}{4} is not allowed. See also A140585. Examples: Let the colon ":" separate two consecutive levels l and l+1. Then n=2 --> 3: {1}{2}, {1}:{2}, {2}:{1}, n=3 --> 10: {1}{2}{3}, {1}{2}:{3}, {3}{1}:{2}, {2}{3}:{1}, {1}:{2}:{3}, {3}:{1}:{2}, {2}:{3}:{1}, {1}:{3}:{2}, {2}:{1}:{3}, {3}:{2}:{1}. - Thomas Wieder, May 17 2008
n identical objects are painted by dipping them into a long row of cans of paint of distinct colors. Begining with the first can and not skipping any cans k, 1<=k<=n, objects are dipped (painted) and not more objects are dipped into any subsequent can than were dipped into the previous can. The painted objects are then linearly ordered. - Geoffrey Critzer, Jun 08 2009
a(n) is the number of partitions of n where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once in the partition. a(3) = 10: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a. - Alois P. Heinz, Aug 30 2015
Also the number of ordered set partitions of {1,...,n} with weakly decreasing block sizes. - Gus Wiseman, Sep 03 2018
The parity of a(n) is that of A000110(A000120(n)), so a(n) is even if and only if A000120(n) == 2 (mod 3). - Álvar Ibeas, Aug 11 2020

			For n=3, say the first three cans in the row contain red, white, and blue paint respectively. The objects can be painted r,r,r or r,r,w or r,w,b and then linearly ordered in 1 + 3 + 6 = 10 ways. - _Geoffrey Critzer_, Jun 08 2009
From _Gus Wiseman_, Sep 03 2018: (Start)
The a(3) = 10 ordered set partitions with weakly decreasing block sizes:
  {{1},{2},{3}}
  {{1},{3},{2}}
  {{2},{1},{3}}
  {{2},{3},{1}}
  {{3},{1},{2}}
  {{3},{2},{1}}
  {{2,3},{1}}
  {{1,2},{3}}
  {{1,3},{2}}
  {{1,2,3}}
(End)

Abramowitz and Stegun, Handbook, p. 831, column labeled "M_1".
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 126.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. E. Hoffman, Updown categories: Generating functions and universal covers, arXiv preprint arXiv:1207.1705 [math.CO], 2012.
A. Knopfmacher, A. M. Odlyzko, B. Pittel, L. B. Richmond, D. Stark, G. Szekeres, and N. C. Wormald, The Asymptotic Number of Set Partitions with Unequal Block Sizes, The Electronic Journal of Combinatorics, 6 (1999), R2.
S. Schreiber & N. J. A. Sloane, Correspondence, 1980.

Main diagonal of: A226873, A261719, A309973.
Row sums of: A226874, A262071, A327803.
Column k=1 of A309951.
Column k=0 of A327801.
Cf. A000041, A000110, A000258, A000670, A007837, A008277, A008480, A036038, A140585, A178682, A212855, A247551, A300335, A318762.

A005651b := proc(k) add( d/(d!)^(k/d),d=numtheory[divisors](k)) ; end proc:
A005651 := proc(n) option remember; local k ; if n <= 1 then 1; else (n-1)!*add(A005651b(k)*procname(n-k)/(n-k)!, k=1..n) ; end if; end proc:
seq(A005651(k), k=0..10) ; # R. J. Mathar, Jan 03 2011
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, n!,
      b(n, i-1) +binomial(n, i)*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 29 2015, Dec 12 2016

Table[Total[n!/Map[Function[n, Apply[Times, n! ]], IntegerPartitions[n]]], {n, 0, 20}] (* Geoffrey Critzer, Jun 08 2009 *)
Table[Total[Apply[Multinomial, IntegerPartitions[n], {1}]], {n, 0, 20}] (* Jean-François Alcover and Olivier Gérard, Sep 11 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[t==1, 1/n!, Sum[b[n-j, j, t-1]/j!, {j, i, n/t}]]; a[n_] := If[n==0, 1, n!*b[n, 0, n]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 20 2015, after Alois P. Heinz *)

a(m,n):=if n=m then 1 else sum(binomial(n,k)*a(k,n-k),k,m,(n/2))+1;
makelist(a(1,n),n,0,17); /* Vladimir Kruchinin, Sep 06 2014 */

a(n)=my(N=n!,s);forpart(x=n,s+=N/prod(i=1,#x,x[i]!));s \\ Charles R Greathouse IV, May 01 2015

{ my(n=25); Vec(serlaplace(prod(k=1, n, 1/(1-x^k/k!) + O(x*x^n)))) } \\ Andrew Howroyd, Dec 20 2017

E.g.f.: 1 / Product (1 - x^k/k!).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} d*d!^(-k/d). - Vladeta Jovovic, Oct 14 2002
a(n) ~ c * n!, where c = Product_{k>=2} 1/(1-1/k!) = A247551 = 2.52947747207915264... . - Vaclav Kotesovec, May 09 2014
a(n) = S(n,1), where S(n,m) = sum(k=m..n/2 , binomial(n,k)*S(n-k,k))+1, S(n,n)=1, S(n,m)=0 for nVladimir Kruchinin, Sep 06 2014
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Jun 18 2018

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

A141199 Number of hierarchical ordered partitions of partitions.

Original entry on oeis.org

1, 1, 3, 7, 17, 38, 87, 191, 421, 911, 1963, 4186, 8885, 18724, 39284, 82005, 170521, 353214, 729290, 1501184, 3081869, 6311404, 12896983, 26301515, 53541702, 108815626, 220824295, 447524559, 905850001, 1831526719

Offset: 0

Thomas Wieder, Jun 13 2008, Jun 29 2008, Jul 28 2008

nonn

Consider the "ordered partitions of partitions" as described in A055887. They are produced by introducing separators (a term used by J. Riordan) between the parts of a partition. If a partition has P parts, then it is possible to introduce 1, 2, ... P-1 separators. Let "|" denote such a separator. We just append 1,2,...,P-1 separators to each integer partition of n and subsequently form all permutation of the resulting list (which is composed of parts and separators).
There are some rules: If we do not append a separator, then we do not perform any permutation. Furthermore, we do not accept permutations which have a dangling separator in front of the integer parts or past them. E.g. the permutations [|,1,2,3] and [1,2,3,|] are forbidden. Furthermore, sequences of separators as "|,|" are forbidden.
Now we impose a further restriction on the permutations. Consider the elements between two separators. We call their number "occupation number". We just request that the occupation number of a ordered partition is monotonically decreasing (if we start from the left to the right of a permutation written in our notation). If we interpret a separator as a level, then we can speak of a hierarchy. E.g. we do not count [1,|,2,3,|,4] as a hierarchy, but we accept [1,2|,3,4] as a hierarchy. We thus speak of "hierarchically ordered partitions of partitions" for this sequence.
With the generating function f := z -> 1/(mul(1-z^i/mul(1-z^j,j=1..i), i=1..25)); we get the asymptotic expansion using the command equivalent (f(z),z,n);
The result is 3.788561346*exp(-n)^(-log(2)) + O(1/n*exp(-n)^(-log(2))). Let fas := n -> 3.788562346*exp(-n)^(-log(2)); then for n=60 we get fas(60)/A141199(60)= .4367915009e19/4344507472742893655 = 1.005387846.
In short, a(n) is the number of finite sequences of integer partitions with weakly decreasing lengths and total sum n. The case of twice-partitions is A358831. A version choosing compositions is A218482. The strictly decreasing case is A358836. For ordered set partitions we have A005651. For weakly decreasing bigomega see A358335. - Gus Wiseman, Dec 05 2022

			n=1:
[1]
-------------------------
n=2:
[1, 1],
[1, "|", 1],
[2]
-------------------------
n=3:
[1, 2],
[1, "|", 1, "|", 1],
[1, 1, 1],
[3],
[2, "|", 1],
[1, 1, "|", 1],
[1, "|", 2]
-------------------------
n=4:
[1, 1, 1, "|", 1],
[1, 1, "|", 1, 1],
[2, 2],
[1, 3],
[1, 1, 1, 1],
[1, 1, 2],
[4],
[1, "|", 1, "|", 1, "|", 1],
[1, 2, "|", 1],
[1, 1, "|", 2],
[1, 1, "|", 1, "|", 1],
[2, "|", 1, "|", 1],
[1, "|", 2, "|", 1],
[1, "|", 1, "|", 2],
[1, "|", 3],
[3, "|", 1],
[2, "|", 2].

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Thomas Wieder, The number of certain rankings and hierarchies formed from labeled or unlabeled elements and sets, Applied Mathematical Sciences, vol. 3, 2009, no. 55, 2707 - 2724. [_Thomas Wieder_, Nov 14 2009]

Cf. A055887, A083355, A140585.

Cf. A000041, A000219, A001970, A005651, A063834, A129519, A218482, A358335, A358831, A358836, A358908.

A Maple program to generate these "hierarchically ordered partitions of partitions" is available on request.
An asymptotic expansion can be found using the generating function given by Vladeta Jovovic. For that purpose we use the Maple program "equivalent" from Bruno Salvy (http://ago.inria.fr/libraries/libraries.html).

my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k/prod(j=1, k, 1-x^j))) \\ Seiichi Manyama, Jan 18 2022

G.f.: 1/Product_{i>=1} (1-x^i/Product_{j=1..i} (1-x^j)). - Vladeta Jovovic, Jul 16 2008

More terms from Vladeta Jovovic, Jul 16 2008

a(0)=1 prepended by Seiichi Manyama, Jan 18 2022

A247551 Decimal expansion of Product_{k>=2} 1/(1-1/k!).

Original entry on oeis.org

2, 5, 2, 9, 4, 7, 7, 4, 7, 2, 0, 7, 9, 1, 5, 2, 6, 4, 8, 1, 8, 0, 1, 1, 6, 1, 5, 4, 2, 5, 3, 9, 5, 4, 2, 4, 1, 1, 7, 8, 7, 0, 2, 3, 9, 4, 8, 4, 5, 9, 9, 7, 3, 3, 7, 5, 8, 4, 9, 3, 4, 9, 8, 2, 5, 0, 0, 2, 1, 1, 8, 7, 8, 0, 0, 8, 6, 6, 9, 9, 1, 2, 1, 9, 9, 8, 8, 3, 6, 4, 6, 2, 5, 2, 6, 1, 4, 9, 5, 5, 1, 6, 4, 3, 2

Offset: 1

Vaclav Kotesovec, Sep 19 2014

			2.5294774720791526481801161542539542411787023948459973375849349825...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 269.
A. Knopfmacher, A. M. Odlyzko, B. Pittel, L. B. Richmond, D. Stark, G. Szekeres, and N. C. Wormald, The Asymptotic Number of Set Partitions with Unequal Block Sizes, The Electronic Journal of Combinatorics, Volume 6 (1999), Research Paper #R2.

Cf. A005651, A035341, A072605, A140585, A183239, A209668, A261600, A282529, A292713, A320566, A327827, A335643.

evalf(1/product(1-1/k!,k=2..infinity),100); # Vaclav Kotesovec, Sep 19 2014

digits = 105;
RealDigits[NProduct[1/(1-1/k!), {k, 2, Infinity}, WorkingPrecision -> digits+10, NProductFactors -> digits], 10, digits][[1]] (* Jean-François Alcover, Nov 02 2020 *)

default(realprecision,150); 1/prodinf(k=2,1 - 1/k!) \\ Vaclav Kotesovec, Sep 21 2014

Product_{k>=2} 1/(1-1/k!).
Equals lim_{n -> oo} A005651(n) / n!.
Equals 1/A282529. - Amiram Eldar, Sep 15 2023

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

A305547 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k/k!).

Original entry on oeis.org

1, 1, 2, 8, 37, 182, 1039, 7149, 56382, 479220, 4280247, 40406984, 410453366, 4539623168, 54431372233, 695801259947, 9312538336475, 128985882874288, 1842668013046405, 27238267120063415, 419396473955088310, 6769168354222927254, 114837651830425810381, 2042782103293394499566

Offset: 0

Ilya Gutkovskiy, Jun 15 2018

nonn

Stirling transform of A007837.

Seiichi Manyama, Table of n, a(n) for n = 0..500
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A007837, A140585, A305987.

b:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
       d=numtheory[divisors](k))*(n-1)!/(n-k)!*b(n-k), k=1..n))
    end:
a:= n-> add(Stirling2(n, k)*b(k), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018

nmax = 23; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) (Exp[x] - 1)^(j k)/((j!)^k 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, -# (-#!)^(-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, 23}]

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*(exp(x) - 1)^(j*k)/((j!)^k*k)).

a(n) = Sum_{k=0..n} Stirling2(n,k)*A007837(k).

A306039 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1 + x)^k/k!).

Original entry on oeis.org

1, 1, 2, 3, 14, 0, 359, -1988, 28706, -312210, 4387572, -62769366, 1006242599, -17203315363, 318393704043, -6296931104285, 133039045075494, -2986262905171914, 71018001954178952, -1783064497977512206, 47133484019671647932, -1308274154275749372040, 38042727898691562357962

Offset: 0

Ilya Gutkovskiy, Jun 17 2018

sign

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A005651, A140585, A306040.

a:=series(mul(1/(1-log(1+x)^k/k!),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[Product[1/(1 - Log[1 + x]^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[Log[1 + x]^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] Total[Apply[Multinomial, IntegerPartitions[k], {1}]], {k, 0, n}], {n, 0, 22}]

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} log(1 + x)^(j*k)/(k*(j!)^k)).

a(n) = Sum_{k=0..n} Stirling1(n,k)*A005651(k).

A345756 E.g.f.: Product_{k>=1} 1/(1 - (exp(x) - 1)^k)^(1/k!).

Original entry on oeis.org

1, 1, 4, 20, 132, 1057, 10036, 110168, 1369395, 19009207, 291638340, 4898978911, 89387432140, 1760380295559, 37222139393757, 841009071062929, 20219172890524757, 515336552717107810, 13879978696592456136, 393920374851547833518, 11749388855614114735431

Offset: 0

Seiichi Manyama, Jun 26 2021

nonn

Stirling transform of A209902.

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A048993, A140585, A209902, A345757, A345758.

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-(exp(x)-1)^k)^(1/k!))))

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, (exp((exp(x)-1)^k)-1)/k))))

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

a(n) = Sum_{k=0..n} Stirling2(n,k) * A209902(k).

A347005 E.g.f.: Product_{k>=1} 1 / (1 - exp(x) * x^k / k!).

Original entry on oeis.org

1, 1, 5, 28, 205, 1856, 19964, 249005, 3535613, 56339884, 996009280, 19350090365, 409850078356, 9400728524669, 232154433941057, 6141705628777193, 173295665869432733, 5195039603196754564, 164890990869273983108, 5524278740902526776085, 194815729875439415542760

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Cf. A005651, A140585, A265953, A347006.

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

E.g.f.: exp( Sum_{k>=1} ( Sum_{d|k} exp(d*x) / (d * ((k/d)!)^d) ) * x^k ).
E.g.f.: Product_{k>=1} 1 / (1 - Sum_{j>=k} binomial(j,k) * x^j / j!).
a(n) ~ c * n! / ((1 + LambertW(1)) * LambertW(1)^n), where c = Product_{k>=2} (1/(1 - LambertW(1)^(k-1)/k!)) = 1.487589725380080111479849424209442083... - Vaclav Kotesovec, Aug 10 2021

A307525 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1/(1 - x))^k/k!).

Original entry on oeis.org

1, 1, 4, 21, 146, 1240, 12479, 144970, 1908682, 28079550, 456458832, 8125189974, 157190542607, 3284222304545, 73705849847317, 1768479436456975, 45180024672023814, 1224529894981726614, 35096983241255523572, 1060703070504583747430, 33714045363258013414692

Offset: 0

Ilya Gutkovskiy, Apr 12 2019

nonn

Cf. A005651, A140585, A306039, A320349.

nmax = 20; CoefficientList[Series[Product[1/(1 - Log[1/(1 - x)]^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[Sum[Log[1/(1 - x)]^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] Total[Apply[Multinomial, IntegerPartitions[k], {1}]], {k, 0, n}], {n, 0, 20}]

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} log(1/(1 - x))^(j*k)/(k*(j!)^k)).
a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A005651(k).
a(n) ~ c * sqrt(2*Pi) * n^(n + 1/2) / (exp(1) - 1)^(n+1), where c = A247551 = Product_{k>=2} 1/(1-1/k!). - Vaclav Kotesovec, Apr 13 2019

cp's OEIS Frontend

A144792 EXP transform of A140585.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A005651 Sum of multinomial coefficients (n_1+n_2+...)!/(n_1!*n_2!*...) where (n_1, n_2, ...) runs over all integer partitions of n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

PARI

Formula

Extensions

A141199 Number of hierarchical ordered partitions of partitions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A247551 Decimal expansion of Product_{k>=2} 1/(1-1/k!).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A305547 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k/k!).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A306039 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1 + x)^k/k!).

Views

Author

A005651 Sum of multinomial coefficients (n_1+n_2+...)!/(n_1!n_2!...) where (n_1, n_2, ...) runs over all integer partitions of n.