A110143 -id:A110143 - cp's OEIS frontend

A279819 Coefficients in asymptotic expansion of sequence A110143.

1, 0, 2, 5, 23, 106, 537, 3143, 20485, 143747, 1078660, 8680687, 74914773, 690204588, 6749661220, 69732043730, 758671016406, 8674112392913, 104037242257509, 1306629414101911, 17148719951169617, 234689253311285406, 3342159005325362828, 49430840838485256475

Offset: 0

Vaclav Kotesovec, Dec 19 2016

nonn

			A110143(n)/n! ~ 1 + 2/n^2 + 5/n^3 + 23/n^4 + 106/n^5 + 537/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..132
Joseph Ben Geloun and Sanjaye Ramgoolam, All-orders asymptotics of tensor model observables from symmetries of restricted partitions, arXiv:2106.01470 [hep-th], Jun 02 2021.

Cf. A110143.

A057005 Number of conjugacy classes of subgroups of index n in free group of rank 2.

Original entry on oeis.org

1, 3, 7, 26, 97, 624, 4163, 34470, 314493, 3202839, 35704007, 433460014, 5687955737, 80257406982, 1211781910755, 19496955286194, 333041104402877, 6019770408287089, 114794574818830735, 2303332664693034476, 48509766592893402121, 1069983257460254131272

Offset: 1

N. J. A. Sloane, Sep 09 2000

nonn

Number of (unlabeled) dessins d'enfants with n edges. A dessin d'enfant ("child's drawing") by A. Grothendieck, 1984, is a connected bipartite multigraph with properly bicolored nodes (w and b) in which a cyclic order of the incident edges is assigned to every node. For n=2 these are w--b--w, b--w--b and w==b. - Valery A. Liskovets, Mar 17 2005
Also (apparently), a(n+1) = number of sensed hypermaps with n darts on a surface of any genus (see Walsh 2012). - N. J. A. Sloane, Aug 01 2012
Response from Timothy R. Walsh, Aug 01 2012: The conjecture in the previous comment is true. A combinatorial map is a connected graph, loops and multiple edges allowed, in which a cyclic order of the incident edge-ends is assigned to every node. The equivalence between combinatorial maps and topological maps was conjectured by several researchers and finally proved by Jones and Singerman. In my 1975 paper "Generating nonisomorphic maps without storing them", I established a genus-preserving bijection between hypermaps with n darts, w vertices and b edges and properly bicolored bipartite maps with n edges, w white vertices and b black vertices.  A bipartite map can't have any loops; so a combinatorial bipartite map is a multigraph and it suffices to impose a cyclic order of the edges, rather than the edge-ends, incident to each node.  Thus it is just the child's drawing described above by Liskovets.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.

Alois P. Heinz, Table of n, a(n) for n = 1..450
M. Deryagina, On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices, Preprint 2016.
J. B. Geloun and S. Ramgoolam, Counting Tensor Model Observables and Branched Covers of the 2-Sphere, arXiv preprint arXiv:1307.6490 [hep-th], 2013.
Paawan Jethva, Exploring the Euler Characteristics of Dessins d'Enfants, Univ. Adelaide (Australia, 2023).
G. A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37:2 (1978), 273-307.
J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Th., 23 (1996), 105-109.
J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
V. A. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Applic. Math., 52 (1998), 91-120.
Carlos I. Pérez-Sánchez, The full Ward-Takahashi Identity for colored tensor models, arXiv preprint arXiv:1608.08134 [math-ph], 2016.
M. Planat, A. Giorgetti, F. Holweck, and M. Saniga, Quantum contextual finite geometries from dessins d'efants, arXiv:1310.4267 [quant-ph], 2013-2015.
P. Vrana, On the algebra of local unitary invariants of pure and mixed quantum states, J. Phys A: Math. Theor. 44 (2011) 225304 doi:10.1088/1751-8113/44/22/225304, Table 2.
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
Timothy R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.
L. Zapponi, What is a dessin d'enfant?, Notices AMS, 50:7, 2003, 788-789.
Index entries for sequences related to groups

Cf. A057004-A057013. Inverse Euler transform of A110143.

f[1] = {a[0] -> 0, a[1] -> 1};
f[max_] := f[max] = (p1 = Product[(1 - x^n)^(-a[n]), {n, 0, max}]; p2 = Product[Sum[j!*If[j == 0, 1, i^j]*x^(i*j), {j, 0, max}], {i, 0, max}];
s = Series[p1 - p2 /. f[max - 1], {x, 0, max}] // Normal // Expand;
sol = Thread[CoefficientList[s, x] == 0] // Solve // First;
Join[f[max - 1], sol]);
Array[a, 22] /. f[22] (* Jean-François Alcover, Mar 11 2014, updated Jan 01 2021 *)

prod_{n>0} (1-x^n)^{-a(n)} = prod_{i>0} sum_{j>=0} j!*i^j*x^{i*j}. (Liskovets) - Valery A. Liskovets, Mar 17 2005 ... and both sides = sum_{j>=0} A110143(j)*x^j . - R. J. Mathar, Oct 18 2012

a(n) ~ n! * (1 - 1/n - 1/n^2 - 4/n^3 - 23/n^4 - 171/n^5 - 1542/n^6 - 16241/n^7 - 194973/n^8 - 2622610/n^9 - 39027573/n^10 - ...), for the coefficients see A113869. - Vaclav Kotesovec, Aug 09 2019

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

A096161 Row sums for triangle A096162.

Original entry on oeis.org

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

Offset: 1

Alford Arnold, Jun 18 2004

nonn

Also, partitions such that a set of k equal terms are labeled 1 through k and can appear in any order. For example, the partition 3+2+2+2+1+1+1+1 of 13 appears 1!*3!*4!=144 times because there are 1! ways to order the one "3," 3! ways to order the three "2"s, ... - Christian G. Bower, Jan 17 2006

			1 1 2 1 3 6 1 4 6 12 24 ... A036038
1 1 1 1 3 1 1 4 3 6 1 ... A036040
1 1 2 1 1 6 1 1 2 2 24 ... A096162
so a(n) begins 1 3 8 30 ... A096161

Seiichi Manyama, Table of n, a(n) for n = 1..449

Cf. A005651, A000110, A036038, A036040, A096162, A110143, A287899.

nmax = 25; Rest[CoefficientList[Series[Product[Sum[k!*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 10 2019 *)
m = 25; Rest[CoefficientList[Series[Product[-Gamma[0, -1/x^j] * Exp[-1/x^j], {j, 1, m}] / x^(m*(m + 1)/2), {x, 0, m}], x]] (* Vaclav Kotesovec, Dec 07 2020 *)

{ my(n=25); Vec(prod(k=1, n, O(x*x^n) + sum(r=0, n\k, x^(r*k)*r!))) }

G.f.: B(x)*B(x^2)*B(x^3)*... where B(x) is g.f. of A000142. - Christian G. Bower, Jan 17 2006
G.f.: Product_{k>0} Sum_{r>=0} x^(r*k)*r!. - Andrew Howroyd, Dec 22 2017
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, Aug 10 2019

More terms from Vladeta Jovovic, Jun 22 2004

A160449 Array read by antidiagonals: T(n,k) is the number of isomorphism classes of n-fold coverings of a connected graph with Betti number k (1 <= n, 0 <= k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 5, 11, 8, 1, 1, 7, 43, 49, 16, 1, 1, 11, 161, 681, 251, 32, 1, 1, 15, 901, 14721, 14491, 1393, 64, 1, 1, 22, 5579, 524137, 1730861, 336465, 8051, 128, 1, 1, 30, 43206, 25471105, 373486525, 207388305, 7997683, 47449, 256, 1

Offset: 0

N. J. A. Sloane, Nov 13 2009

T(n,k) is the number of orbits of the conjugacy action of Sym(n) on Sym(n)^k [Kwak and Lee, 2001]. - Álvar Ibeas, Mar 25 2015

			The array begins:
      k=0   k=1   k=2   k=3     k=4       k=5
  n=1   1     1     1     1       1         1
  n=2   1     2     4     8      16        32
  n=3   1     3    11    49     251      1393
  n=4   1     5    43   681   14491    336465
  n=5   1     7   161 14721 1730861 207388305

Álvar Ibeas, Table of n, a(n) for n = 0..1829
Michael W. Hero and Jeb F. Willenbring, Stable Hilbert series as related to the measurement of quantum entanglement, Discrete Mathematics, 309 (2009), 6508-6514. See Table 3.
J. H. Kwak and J. Lee, Isomorphism classes of graph bundles. Can. J. Math., 42(4), 1990, pp. 747-761.
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. See Table 2.

Rows: A000012, A000079, A074528, A160450, A160454, A176709.
Columns: A000041, A110143, A152612, A160446, A160447, A160448.
Cf. A057004.

def A160449(n, k):
    return sum(p.aut()**(k - 1) for p in Partitions(n))
# Álvar Ibeas, Mar 25 2015

Name clarified and more terms added by Álvar Ibeas, Mar 25 2015

A110141 Triangle, read by rows, where row n 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).

Original entry on oeis.org

1, 1, 2, 2, 6, 2, 3, 24, 4, 3, 8, 4, 120, 12, 6, 8, 4, 6, 5, 720, 48, 18, 16, 8, 6, 5, 48, 8, 18, 6, 5040, 240, 72, 48, 24, 12, 10, 48, 8, 18, 6, 24, 10, 12, 7, 40320, 1440, 360, 192, 96, 36, 30, 96, 16, 36, 12, 24, 10, 12, 7, 384, 32, 36, 12, 15, 32, 8, 362880, 10080, 2160, 960

Offset: 0

Paul D. Hanna, Jul 13 2005

Row n starts with n!, after which the following pattern holds. When terms of row n are divided by a list of factorials, with (n-j-1)! repeated A002865(j+1) times in the list as j=1..n-1, the result is the initial terms of A110142. E.g., row 6 is: {720,48,18,16,8,6,5,48,8,18,6}; divide by respective factorials: {6!,4!,3!,2!,2!,1!,1!,0!,0!,0!,0!} with {4!,3!,2!,1!,0!} respectively occurring {1,1,2,2,4} times (A002865), yields the initial terms of A110142: {1,2,3,8,4,6,5,48,8,18,6}.
The term of the sequence corresponding to the product c_1^{n_1}c_2^{n_2}...c_k^{n_k} is equal to the number of elements in the centralizer of a permutation of n_1+2n_2+...+kn_k elements whose cycle type is 1^{n_1}2^{n_2}...k^{n^k}. (This fact is very standard, in particular, for the theory of symmetric functions.) - Vladimir Dotsenko, Apr 19 2009
Multiplying the values of row n by the corresponding values in row n of A102189, one obtains n!. - Jaimal Ichharam, Aug 06 2015
a(n,k) is the number of permutations in S_n that commute with a permutation having cycle type "k".  Here, the cycle type of an n-permutation pi is the vector (i_1,...,i_n) where i_j is the number of cycles in pi of length j.  These A000041(n) vectors can be ordered in reverse lexicographic order.  The k-th cycle type is the k-th vector in this ordering. - Geoffrey Critzer, Jan 18 2019

			Coefficients [x^n] exp(c1*x + (c2/2)*x^2 + (c3/3)*x^3 + ...) begin:
[x^0]: 1;
[x^1]: 1*c1;
[x^2]: (1/2)*c1^2 + (1/2)*c2;
[x^3]: (1/6)*c1^3 + (1/2)*c1*c2 + (1/3)*c3;
[x^4]: (1/24)*c1^4 + (1/4)*c1^2*c2 + (1/3)*c1*c3 + (1/8)*c2^2 + (1/4)*c4;
[x^5]: (1/120)*c1^5 + (1/12)*c1^3*c2 + (1/6)*c1^2*c3 + (1/8)*c1*c2^2 + (1/4)*c1*c4 + (1/6)*c2*c3 + (1/5)*c5;
[x^6]: (1/720)*c1^6 + (1/48)*c1^4*c2 + (1/18)*c1^3*c3 + (1/16)*c1^2*c2^2 + (1/8)*c1^2*c4 + (1/6)*c1*c2*c3 + (1/5)*c1*c5 + (1/48)*c2^3 + (1/8)*c2*c4 + (1/18)*c3^2 + (1/6)*c6;
forming this triangle of unit fraction coefficients:
1;
1;
2,2;
6,2,3;
24,4,3,8,4;
120,12,6,8,4,6,5;
720,48,18,16,8,6,5,48,8,18,6;
5040,240,72,48,24,12,10,48,8,18,6,24,10,12,7;
40320,1440,360,192,96,36,30,96,16,36,12,24,10,12,7,384,32,36,12,15,32,8;
362880,10080,2160,960,480,144,120,288,48,108,36,48,20,24,14,384,32,36,12,15,32,8,144,40,24,14,162,18,20,9; ...

Macdonald, I. G. Symmetric functions and Hall polynomials. Oxford University Press, 1995. [From Vladimir Dotsenko, Apr 19 2009]

Cf. A000041, A002865, A102189, A110142, A110143 (row sums).

First, second and third entries of each row are given (up to an offset) by A000142, A052849, and A052560 respectively. - Vladimir Dotsenko, Apr 19 2009

Table[n!/CoefficientRules[n! CycleIndex[SymmetricGroup[n], s]][[All, 2]], {n, 1, 8}] // Grid (* Geoffrey Critzer, Jan 18 2019 *)

Number of terms in row n is A000041(n) (partition numbers). The unit fractions of each row sum to unity: Sum_{k=1..A000041(n)} 1/T(n, k) = 1.

a(n,k) = n!/A181897(n,k). - Geoffrey Critzer, Jan 18 2019

A326985 G.f.: B(x)B(2x^2)B(3x^3)*..., where B(x) is g.f. of A000312.

Original entry on oeis.org

1, 1, 6, 32, 287, 3222, 47606, 831488, 16890792, 389286222, 10037183606, 286154919078, 8937624574652, 303483905672078, 11130904101218094, 438532313635906858, 18470060947222927499, 828155619735377936654, 39384843256547964375436, 1980138439071577626157382

Offset: 0

Vaclav Kotesovec, Aug 10 2019

nonn

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

Cf. A000312, A110143, A326986.

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)*i^j*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*j^k*x^(j*k), {k, 1, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ n^n.

A271708 Triangle read by rows, T(n,k) = Sum_{p in P(n,k)} Aut(p) where P(n,k) are the partitions of n with largest part k and Aut(p) = 1^j[1]j[1]!...n^j[n]j[n]! where j[m] is the number of parts in the partition p equal to m; for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 6, 2, 3, 0, 24, 12, 3, 4, 0, 120, 20, 12, 4, 5, 0, 720, 112, 42, 16, 5, 6, 0, 5040, 336, 126, 44, 20, 6, 7, 0, 40320, 2112, 492, 188, 55, 24, 7, 8, 0, 362880, 11712, 2802, 640, 215, 66, 28, 8, 9, 0, 3628800, 92160, 16938, 3624, 830, 258, 77, 32, 9, 10

Offset: 0

Peter Luschny, Apr 17 2016

Also T(n,k) = Sum_{p in P(n,k)} Cen(p) where Cen(p) is the size of the centralizer of any permutation of cycle type p.

			Triangle starts:
[1]
[0, 1]
[0, 2, 2]
[0, 6, 2, 3]
[0, 24, 12, 3, 4]
[0, 120, 20, 12, 4, 5]
[0, 720, 112, 42, 16, 5, 6]
[0, 5040, 336, 126, 44, 20, 6, 7]
[0, 40320, 2112, 492, 188, 55, 24, 7, 8]

Cf. A110143 (row sums), A126074.

def A271708(n,k):
    P = Partitions(n, max_part=k, inner=[k])
    return sum([p.aut() for p in P])
for n in (0..9): print([A271708(n,k) for k in (0..n)])

A309618 a(n) = Sum_{k=0..floor(n/2)} k! * 2^k * (n - 2*k)!.

Original entry on oeis.org

1, 1, 4, 8, 36, 140, 832, 5376, 42432, 374592, 3720960, 40694784, 486679296, 6310114560, 88168366080, 1320468480000, 21101183631360, 358354687426560, 6444941507297280, 122367252835860480, 2445878526994022400, 51337143210820239360, 1128918790687649955840

Offset: 0

Vaclav Kotesovec, Aug 10 2019

nonn

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

Cf. A000142, A003149, A107713, A110143, A309619.

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

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

a(n) ~ n! * (1 + 2/n^2 + 2/n^3 + 10/n^4 + 50/n^5 + 250/n^6 + 1442/n^7 + 9514/n^8 + 68882/n^9 + 539098/n^10 + ...), for coefficients see A326983.

A327014 Number of orbits of Sym(n)^2 where Sym(n) acts by conjugation such that both permutations in a representative pair have the same cycle type.

Original entry on oeis.org

1, 1, 2, 5, 13, 35, 135, 613, 3624, 25230, 203640, 1842350, 18535683, 204650313

Offset: 0

Derek Lim, Aug 13 2019

			For n = 2, representatives of the a(2) = 2 orbits are: (e,e), ((12), (12)), where e is identity.

Amit Harlev, Charles R. Johnson, and Derek Lim, The Doubly Stochastic Single Eigenvalue Problem: A Computational Approach, arXiv:1908.03647 [math.SP], 2019.

Cf. A110143, A327015.

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

cp's OEIS Frontend

A279819 Coefficients in asymptotic expansion of sequence A110143.

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A057005 Number of conjugacy classes of subgroups of index n in free group of rank 2.

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A096161 Row sums for triangle A096162.

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A160449 Array read by antidiagonals: T(n,k) is the number of isomorphism classes of n-fold coverings of a connected graph with Betti number k (1 <= n, 0 <= k).

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A110141 Triangle, read by rows, where row n 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).

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A326985 G.f.: B(x)*B(2*x^2)*B(3*x^3)*..., where B(x) is g.f. of A000312.

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A271708 Triangle read by rows, T(n,k) = Sum_{p in P(n,k)} Aut(p) where P(n,k) are the partitions of n with largest part k and Aut(p) = 1^j[1]*j[1]!*...*n^j[n]*j[n]! where j[m] is the number of parts in the partition p equal to m; for n>=0 and 0<=k<=n.

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A309618 a(n) = Sum_{k=0..floor(n/2)} k! * 2^k * (n - 2*k)!.

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A326985 G.f.: B(x)B(2x^2)B(3x^3)*..., where B(x) is g.f. of A000312.

A271708 Triangle read by rows, T(n,k) = Sum_{p in P(n,k)} Aut(p) where P(n,k) are the partitions of n with largest part k and Aut(p) = 1^j[1]j[1]!...n^j[n]j[n]! where j[m] is the number of parts in the partition p equal to m; for n>=0 and 0<=k<=n.

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