A096161 -id:A096161 - cp's OEIS frontend

A293266 Coefficients in asymptotic expansion of sequence A096161 and A161779.

1, 0, 1, 2, 7, 28, 121, 587, 3205, 19201, 123684, 850873, 6248839, 48948805, 407666212, 3594074850, 33405529547, 326310068618, 3342124657507, 35827145094057, 401325346421766, 4689964771177970, 57081316456694665, 722295766109273335, 9486188532177356598

Offset: 0

Vaclav Kotesovec, Oct 04 2017

nonn

			A096161(n) / n! ~ 1 + 1/n^2 + 2/n^3 + 7/n^4 + 28/n^5 + 121/n^6 + ...
A161779(n) / n! ~ 1 + 1/n^2 + 2/n^3 + 7/n^4 + 28/n^5 + 121/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..122

Cf. A096161, A161779.

A110143 Row sums of triangle A110141.

Original entry on oeis.org

1, 1, 4, 11, 43, 161, 901, 5579, 43206, 378360, 3742738, 40853520, 488029621, 6323154547, 88308425755, 1322120265238, 21122364398761, 358647945023885, 6449299885654827, 122436442904193940, 2447046870232798369, 51358050784584629338, 1129314001779283063606

Offset: 0

Paul D. Hanna, Jul 14 2005

nonn

Row n of triangle A110141 lists the denominators of unit fraction coefficients of the products of {c_k}, in ascending order by indices of {c_k}, in the coefficient of x^n in exp(Sum_{k>=1} c_k/k*x^k). There are A000041(n) terms in row n of triangle A110141.
Also, number of orbits of Sym(n)^2 where Sym_n acts by conjugation.  Compare the MathOverflow discussion, also Bogaerts-Dukes 2014, and A241584, A241585. - Peter J. Dukes, May 12 2014
Number of isomorphism classes of n-fold coverings of a connected graph with circuit rank 2 [Kwak and Lee]. - Álvar Ibeas, Mar 25 2015

P. A. MacMahon, The expansion of determinants and permanents in terms of symmetric functions, in Proc. ICM Toronto (ed. J. C. Fields), Toronto University Press, 1924, vol 1, 319-330.
J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. Appears to contain this sequence in Table 2. [Added by N. J. A. Sloane, Nov 12 2009]

Alois P. Heinz, Table of n, a(n) for n = 0..450
Mathieu Bogaerts and Peter Dukes, Semidefinite programming for permutation codes, Discrete Math. 326 (2014), 34--43. MR3188985.
Nicholas Dub, Enumeration of triangulations modulo symmetries and of rooted triangulations counted by their number of (d - 2)-simplices in dimension d ≥ 2, tel-03641958 [cs.OH], Université Paris-Nord - Paris XIII, 2021.
J. B. Geloun, S. Ramgoolam, Counting Tensor Model Observables and Branched Covers of the 2-Sphere, arXiv:1307.6490 [hep-th], 2013.
Joseph Ben Geloun, Sanjaye Ramgoolam, All-orders asymptotics of tensor model observables from symmetries of restricted partitions, arXiv:2106.01470 [hep-th], Jun 02 2021.
J. H. Kwak and J. Lee, Isomorphism classes of graph bundles. Can. J. Math., 42(4), 1990, pp. 747-761.
MathOverflow, A general formula for the number of conjugacy classes of S_n×S_n acted on by S_n [From Peter Dukes, May 12 2014]
Igor Pak, Greta Panova, Damir Yeliussizov, On the largest Kronecker and Littlewood-Richardson coefficients, arXiv:1804.04693 [math.CO], 2018.

Cf. A110141, A000041, A057005, A096161, A241584, A241585, A279819.

Third column of A160449.

# Using a function from Alois P. Heinz in A279038:
b:= proc(n, i) option remember; `if`(n=0, [1],
      `if`(i<1, [], [seq(map(x-> x*i^j*j!,
      b(n-i*j, i-1))[], j=0..n/i)]))
    end:
seq(add(i, i=b(n$2)), n=0..22); # Peter Luschny, Dec 19 2016

Table[Total[Apply[Times, Tally[#]/.{a_Integer,b_}->a^b b!]& /@ IntegerPartitions[n]],{n,0,21}] (* Wouter Meeussen, Oct 17 2014 *)
b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i < 1, {}, Flatten[ Table[ Map[ #*i^j*j!&, b[n-i*j, i-1]], {j, 0, n/i}]]]]; Table[Sum[i, {i, b[n, n]}], {n, 0, 22}] (* Jean-François Alcover, Jul 10 2017, after Alois P. Heinz *)
nmax = 25; CoefficientList[Series[Product[Sum[k!*j^k*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 08 2019 *)
m = 30; CoefficientList[Series[Product[-Gamma[0, -1/(x^j*j)] * Exp[-1/(x^j*j)], {j, 1, m}] / (x^(m*(m + 1)/2)*m!), {x, 0, m}], x] (* Vaclav Kotesovec, Dec 07 2020 *)

def A110143(n):
    return sum(p.aut() for p in Partitions(n))
[A110143(n) for n in range(9)]
# Álvar Ibeas, Mar 26 2015

G.f.: B(x)*B(2*x^2)*B(3*x^3)*..., where B(x) is g.f. of A000142. - Vladeta Jovovic, Feb 18 2007

a(n) ~ n! * (1 + 2/n^2 + 5/n^3 + 23/n^4 + 106/n^5 + 537/n^6 + 3143/n^7 + 20485/n^8 + 143747/n^9 + 1078660/n^10), for coefficients see A279819. - Vaclav Kotesovec, Mar 16 2015

A096162 Let n be a number partitioned as n = b_1 + 2b_2 + ... + nb_n; then T(n) = (b_1)! * (b_2)! * ... (b_n)!. Irregular triangle read by rows, T(n, k) for n >= 1 and 1 <= k <= A000041(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 6, 1, 1, 2, 2, 24, 1, 1, 1, 2, 2, 6, 120, 1, 1, 1, 2, 2, 1, 6, 6, 4, 24, 720, 1, 1, 1, 1, 2, 1, 2, 2, 6, 2, 6, 24, 12, 120, 5040, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 6, 2, 4, 2, 24, 24, 6, 12, 120, 48, 720, 40320, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 6, 6, 2, 2, 2, 2, 6, 24, 6, 12, 4, 24, 120

Offset: 1

Alford Arnold, Jun 20 2004

The partitions of number n are grouped by increasing length and in reverse lexical order for partitions of the same length.

This sequence is in the Abramowitz-Stegun ordering, see A036036. - Hartmut F. W. Hoft, Apr 25 2015

			Illustrating the formula:
1 1 2 1 3 6 1 4 6 12 24 ... A036038
1 1 1 1 3 1 1 4 3  6  1 ... A036040
so
1 1 2 1 1 6 1 1 2  2 24 ... this sequence.
.
From _Hartmut F. W. Hoft_, Apr 25 2015: (Start)
The sequence as a structured triangle. The column headings indicate the number of elements in the underlying partitions. Brackets indicate groups of the products of factorials for all partitions of the same length when there is more than one partition.
     1   2        3        4     5    6
1:   1
2:   1   2
3:   1   1        6
4:   1  [1 2]     2       24
5:   1  [1 1]    [2 2]     6    120
6:   1  [1 1 2]  [2 1 6]  [6 4]  24  720
The partitions, their multiplicities and factorial products associated with the five entries in row n = 4 are:
partitions:         {4}, [{3, 1}, {2, 2}], {2, 1, 1}, {1, 1, 1, 1}
multiplicities:      1,  [{1, 1},  2],     {1, 2},     4
factorial products:  1!, [1!*1!, 2!],      1!*2!,      4!
(End)

Abramowitz and Stegun, Handbook of Mathematical Functions, p. 831, column "M_1" divided by "M_3."

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].

Row sums in A096161.
Row lengths in A000041.
Cf. A036038, A036040, A130561.

(* function a096162[ ] computes complete rows of the triangle *)
row[n_] := Map[Apply[Times, Map[Factorial, Last[Transpose[Tally[#]]]]]&, GatherBy[IntegerPartitions[n], Length], {2}]
triangle[n_] := Map[row, Range[n]]
a096162[n_] := Flatten[triangle[n]]
Take[a096162[9],90] (* data *)  (*Hartmut F. W. Hoft, Apr 25 2015 *)

from collections import Counter
def A096162_row(n):
    h = lambda p: product(map(factorial, Counter(p).values()))
    return [h(p) for k in (0..n) for p in Partitions(n, length=k)]
for n in (1..9): print(A096162_row(n)) # Peter Luschny, Nov 01 2019

T(n, k) = A036038(n,k) / A036040(n,k).

Appears to be n! / A130561(n); e.g., 4! / (24,24,12,12,1) = (1,1,2,2,24). - Tom Copeland, Nov 12 2017

Edited and extended by Christian G. Bower, Jan 17 2006

A120774 Number of ordered set partitions of [n] where equal-sized blocks are ordered with increasing least elements.

Original entry on oeis.org

1, 1, 2, 8, 31, 147, 899, 5777, 41024, 322488, 2749325, 25118777, 245389896, 2554780438, 28009868787, 323746545433, 3933023224691, 49924332801387, 661988844566017, 9138403573970063, 131043199040556235, 1949750421507432009, 30031656711776544610

Offset: 0

Alford Arnold, Jul 12 2006

Old name was: Row sums of A179233.

a(n) is the number of ways to linearly order the blocks in each set partition of {1,2,...,n} where two blocks are considered identical if they have the same number of elements. - Geoffrey Critzer, Sep 29 2011

			A179233 begins 1; 1; 1 1; 6 1 1; 8 3 18 1 1 ... with row sums 1, 1 2 8 31 147 ...
a(3) = 8: 123, 1|23, 23|1, 2|13, 13|2, 3|12, 12|3, 1|2|3. - _Alois P. Heinz_, Apr 27 2017

Alois P. Heinz, Table of n, a(n) for n = 0..525

Cf. A000041, A032011, A049019, A096161, A096162, A196301.
Row sums of A179233, A285824.
Main diagonal of A327244.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      (p+n)!/n!, add(b(n-i*j, i-1, p+j)*combinat
      [multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 27 2017

f[{x_,y_}]:= x!^y y!;   Table[Total[Table[n!,{PartitionsP[n]}]/Apply[Times,Map[f,Map[Tally,Partitions[n]],{2}],2] * Apply[Multinomial,Map[Last,Map[Tally,Partitions[n]],{2}],2]],{n,0,20}]  (* Geoffrey Critzer, Sep 29 2011 *)

Leading 1 inserted, definition simplified by R. J. Mathar, Sep 28 2011

a(15) corrected, more terms, and new name (using Geoffrey Critzer's comment) from Alois P. Heinz, Apr 27 2017

A161779 The sequence of factorials convolved with all its regularly "aerated" variants.

Original entry on oeis.org

1, 1, 3, 8, 30, 133, 768, 5221, 41302, 369170, 3677058, 40338310, 483134179, 6271796072, 87709287104, 1314511438945, 21017751750506, 357102350816602, 6424883282375340, 122025874117476166, 2439726373093186274, 51220112287152570828, 1126575412217509969515

Offset: 0

Gary W. Adamson, Jun 19 2009

nonn

Essentially a duplicate of A096161: 1, followed by A096161.

Convolve A000142 = 1,1,2,6,24,... with 1,0,1,0,2,0,6,0,24,.. and with 1,0,0,1,0,0,2,0,0,6,0,0,24,0,0,.. and with 1,0,0,0,1,0,0,0,2,0,0,0,6,... etc.

			Let the partial products = a, a*b, a*b*c,..., with the first few rows =
(1, 1, 2, 6, 24, 120,...) = a
(1, 1, 3, 7, 28, 128,...) = a*b
(1, 1, 3, 8, 29, 131,...) = a*b*c
(1, 1, 3, 8, 30, 132,...) = a*b*c*d
...converging to A161779

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A096161, row sums of A333144.

read("transforms3") ; read("transforms") ; A161779 := proc(N) local a000142,res,n,j ; a000142 := [seq(n!,n=0..N)] ; res := [seq(op(n,a000142),n=1..N)] ; for j from 1 to N do res := CONV( res, AERATE(a000142,j)) ; od: [seq(op(n,res),n=1..N)] end: A161779(30) ; # R. J. Mathar, Jun 23 2009
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, n!,
      add(b(n-i*j, i-1)*j!, j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 03 2018, revised, Mar 05 2024

b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i] == 0, (n/i)!, 0] + Sum[j! b[n - i j, i + 1], {j, 0, n/i}]];
a[n_] := If[n == 0, 1, b[n, 1]];
a /@ Range[0, 25] (* Jean-François Alcover, Feb 04 2020, after Alois P. Heinz *)

a(n) = A096161(n) for n >= 1. - R. J. Mathar, Jun 26 2009

a(n) ~ n! * (1 + 1/n^2 + 2/n^3 + 7/n^4 + 28/n^5 + 121/n^6 + 587/n^7 + 3205/n^8 + 19201/n^9 + 123684/n^10), for coefficients see A293266. - Vaclav Kotesovec, Oct 04 2017

Extended by R. J. Mathar, Jun 23 2009

A309619 a(n) = Sum_{k=0..floor(n/2)} k! * (n - 2*k)!.

Original entry on oeis.org

1, 1, 3, 7, 28, 128, 754, 5178, 41124, 368220, 3670872, 40290744, 482716896, 6267697920, 87664818960, 1313983544400, 21010949076960, 357007805477280, 6423473819220480, 122003441554176000, 2439346762501367040, 51213306647556506880, 1126446562222595147520

Offset: 0

Vaclav Kotesovec, Aug 10 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..448

Cf. A000142, A003149, A096161, A107713, A309618, A339515.

nmax = 25; CoefficientList[Series[Sum[k!*x^k, {k, 0, nmax}]*Sum[k!*x^(2*k), {k, 0, nmax}], {x, 0, nmax}], x]
Table[Sum[k!*(n-2*k)!, {k, 0, Floor[n/2]}], {n, 0, 25}]

a(n) = sum(k=0, n\2, k! * (n - 2*k)!); \\ Michel Marcus, Dec 08 2020

G.f.: B(x)*B(x^2), where B(x) is g.f. of A000142.

a(n) ~ n! * (1 + 1/n^2 + 1/n^3 + 3/n^4 + 13/n^5 + 57/n^6 + 271/n^7 + 1467/n^8 + 8905/n^9 + 58965/n^10 + ...), for coefficients see A326984.

A326986 G.f.: B(x)B(x^2)B(x^3)*..., where B(x) is g.f. of A000312.

Original entry on oeis.org

1, 1, 5, 29, 266, 3163, 46994, 827107, 16828741, 388308078, 10017853262, 285720195351, 8926575094978, 303172417424680, 11121259586618456, 438207141286916539, 18458204444260001120, 827690809585441201775, 39365349178064541861252, 1979267564496263599093676

Offset: 0

Vaclav Kotesovec, Aug 10 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A000312, A096161, A309652, A326985.

B:= proc(n) option remember; n^n end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1,
      B(n), add(b(j, 1)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 23 2019

nmax = 20; CoefficientList[Series[Product[1+Sum[k^k*x^(j*k), {k, 1, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ n^n.

A126787 G.f.: B(x)B(2!x^2)B(3!x^3)*..., where B(x) is g.f. of A000142.

Original entry on oeis.org

1, 1, 4, 14, 66, 308, 1888, 12240, 95640, 827904, 8106960, 87387264, 1035645312, 13316300928, 184988692800, 2756878875648, 43888205438208, 742943286892800, 13326434312808960, 252448071959572992, 5036116692383428608, 105523926692032447488

Offset: 0

Vladeta Jovovic, Feb 18 2007

Take each Ferrers diagram of the partitions of n, label(linearly order) the dots within each row, then linearly order any of the rows that are of equal length. - Geoffrey Critzer, Mar 21 2009

Alois P. Heinz, Table of n, a(n) for n = 0..450 (terms n=176..300 from Vaclav Kotesovec)

Cf. A096161, A110143.

B:= proc(n) option remember; local x; unapply(`if`(n<=0, 1, B(n-1)(x)+ n! *x^n), x) end: BB:= proc(n) local x, d; unapply(convert(series(mul(B(floor(n/d))(d!*x^d), d=1..n), x, n+1), polynom), x) end: a:= n-> coeff(BB(n)(x), x, n): seq(a(n), n=0..25); # Alois P. Heinz, Sep 25 2008
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, n!,
      add(b(n-i*j, i-1)*j!*i!^j, j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 02 2017

CoefficientList[Series[Product[Sum[x^(n*k) n!^k*k!, {k, 0, 20}], {n, 1, 20}], {x, 0, 20}], x] (* Geoffrey Critzer, Mar 21 2009 *)

a(n) ~ 2*n! * (1 + 1/(2*n) + 3/n^2 + 13/n^3 + 82/n^4 + 587/n^5 + 4966/n^6). - Vaclav Kotesovec, Mar 16 2015

More terms from Alois P. Heinz, Sep 25 2008

A179233 Irregular triangle T(n,k) = A049019(n,k)/A096162(n,k) read along rows, 1<=k <= A000041(n).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 8, 3, 18, 1, 1, 10, 20, 30, 45, 40, 1, 1, 12, 30, 10, 45, 360, 15, 80, 270, 75, 1, 1, 14, 42, 70, 63, 630, 210, 315, 140, 2520, 420, 175, 1050, 126, 1, 1, 16, 56, 112, 35, 84, 1008, 1680, 630, 840, 224, 5040, 1680

Offset: 1

Alford Arnold, Jul 08 2010

Each row n of A049019, of A096162 and of the triangle here has A000041(n) entries.

			A049019(.,.) begins 1; 1; 2, 1; 6, 6, 1; 8, 6, 36, 24, ...
A096162(.,.) begins 1; 1; 2, 1; 1, 6, 1; 1, 2, 2, 24 ...
so
T(.,.) begins ..... 1; 1; 1, 1; 6, 1, 1; 8, 3, 18, 1 ...

Cf. A000041, A000670, A096161.

T(n,k) = A049019(n,k) / A096162(n,k) = A048996(n,k) * A036040(n,k).

Sum_{k=1..A000041(n)} T(n,k) = A120774(n).

Extended, and bivariate indices restored - R. J. Mathar, Jul 13 2010

A179236 Irregular triangle T(n,k) = A096162(n,k)* A036040(n,k)* A048996(n,k)A098546(n,k) A178886(n,k) read by rows, 1<=k<=A000041(n).

Original entry on oeis.org

1, 2, 2, 6, 36, 6, 24, 192, 72, 432, 24, 120, 1200, 2400, 3600, 5400, 4800, 120, 720, 8640, 21600, 7200, 32400, 259200, 10800, 57600, 194400, 54000, 720, 5040, 70560, 211680, 352800, 317520, 3175200, 1058400, 1587600, 705600, 12700800, 2116800, 882000, 5292000, 635040, 5040, 40320, 645120, 2257920, 4515840

Offset: 1

Alford Arnold, Jul 04 2010

			The factor sequences begin
1..1..2..1..1..6
1..1..1..1..3..1
1..1..1..1..2..1
1..2..1..3..3..1
1..1..1..2..2..1
so the present sequence begins
1..2..2..6..36..6

Cf. A000041 (row lengths) A096161 A000110 A000079 A098545 A000522 A179235 (row sums)

cp's OEIS Frontend

A293266 Coefficients in asymptotic expansion of sequence A096161 and A161779.

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A110143 Row sums of triangle A110141.

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A096162 Let n be a number partitioned as n = b_1 + 2*b_2 + ... + n*b_n; then T(n) = (b_1)! * (b_2)! * ... (b_n)!. Irregular triangle read by rows, T(n, k) for n >= 1 and 1 <= k <= A000041(n).

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A120774 Number of ordered set partitions of [n] where equal-sized blocks are ordered with increasing least elements.

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Maple

Mathematica

Extensions

A161779 The sequence of factorials convolved with all its regularly "aerated" variants.

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Mathematica

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Extensions

A309619 a(n) = Sum_{k=0..floor(n/2)} k! * (n - 2*k)!.

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A326986 G.f.: B(x)*B(x^2)*B(x^3)*..., where B(x) is g.f. of A000312.

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A096162 Let n be a number partitioned as n = b_1 + 2b_2 + ... + nb_n; then T(n) = (b_1)! * (b_2)! * ... (b_n)!. Irregular triangle read by rows, T(n, k) for n >= 1 and 1 <= k <= A000041(n).

A326986 G.f.: B(x)B(x^2)B(x^3)*..., where B(x) is g.f. of A000312.

A126787 G.f.: B(x)B(2!x^2)B(3!x^3)*..., where B(x) is g.f. of A000142.

A179236 Irregular triangle T(n,k) = A096162(n,k)* A036040(n,k)* A048996(n,k)A098546(n,k) A178886(n,k) read by rows, 1<=k<=A000041(n).