A113869 -id:A113869 - cp's OEIS frontend

A114038 Analog of A113869 for three generators.

1, 0, -1, 0, -3, -6, -38, -186, -1181, -8094, -61865, -516702, -4688020, -45887352, -481954769, -5406249972, -64506680939, -815807306442, -10901200843386, -153475188129114, -2270769144678657, -35226976789341426, -571781884343282417, -9691701188493783546

Offset: 0

N. J. A. Sloane, Feb 01 2006

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..400
John D. Dixon, Asymptotics of Generating the Symmetric and Alternating Groups, Electronic Journal of Combinatorics, Item R56 of Volume 12(1), 2005.

Related to A113871 in the same way that A113869 is related to A003319.

nmax=30; A113871 = Rest[CoefficientList[Series[1/Sum[(k!)^2 x^k,{k,0,nmax}],{x,0,nmax}],x]]; Table[SeriesCoefficient[1 + Sum[A113871[[j]]/Product[n-i+1,{i,1,j}]^2,{j,1,nmax}],{n,Infinity,k}],{k,0,nmax}] (* Vaclav Kotesovec, Jul 28 2015 *)

a(n) ~ -Pi * n^(n+1) / (2^(n+4) * exp(n) * (log(2))^(n+3/2)). - Vaclav Kotesovec, Jul 28 2015

Missing a(3)=0 and more terms added by Vaclav Kotesovec, Jul 28 2015

A003319 Number of connected permutations of [1..n] (those not fixing [1..j] for 0 < j < n). Also called indecomposable permutations, or irreducible permutations.

Original entry on oeis.org

1, 1, 1, 3, 13, 71, 461, 3447, 29093, 273343, 2829325, 31998903, 392743957, 5201061455, 73943424413, 1123596277863, 18176728317413, 311951144828863, 5661698774848621, 108355864447215063, 2181096921557783605, 46066653228356851631, 1018705098450570562877

Offset: 0

N. J. A. Sloane

Also the number of permutations with no global descents, as introduced by Aguiar and Sottile [Corollaries 6.3, 6.4 and Remark 6.5].
Also the dimensions of the homogeneous components of the space of primitive elements of the Malvenuto-Reutenauer Hopf algebra of permutations. This result, due to Poirier and Reutenauer [Theoreme 2.1] is stated in this form in the work of Aguiar and Sottile [Corollary 6.3] and also in the work of Duchamp, Hivert and Thibon [Section 3.3].
Related to number of subgroups of index n-1 in free group of rank 2 (i.e., maximal number of subgroups of index n-1 in any 2-generator group). See Problem 5.13(b) in Stanley's Enumerative Combinatorics, Vol. 2.
Also the left border of triangle A144107, with row sums = n!. - Gary W. Adamson, Sep 11 2008
Hankel transform is A059332. Hankel transform of aerated sequence is A137704(n+1). - Paul Barry, Oct 07 2008
For every n, a(n+1) is also the moment of order n for the probability density function rho(x) = exp(x)/(Ei(1,-x)*(Ei(1,-x) + 2*I*Pi)) on the interval 0..infinity, with Ei the exponential-integral function. - Groux Roland, Jan 16 2009
Also (apparently), a(n+1) is the number of rooted hypermaps with n darts on a surface of any genus (see Walsh 2012). - N. J. A. Sloane, Aug 01 2012
Also recurrent sequence A233824 (for n > 0) in Panaitopol's formula for pi(x), the number of primes <= x. - Jonathan Sondow, Dec 19 2013
Also the number of mobiles (cyclic rooted trees) with an arrow from each internal vertex to a descendant of that vertex. - Brad R. Jones, Sep 12 2014
Up to sign, Möbius numbers of the shard intersection orders of type A, see Theorem 1.3 in Reading reference. - F. Chapoton, Apr 29 2015
Also, a(n) is the number of distinct leaf matrices of complete non-ambiguous trees of size n. - Daniel Chen, Oct 23 2022

			G.f. = 1 + x + x^2 + 3*x^3 + 13*x^4 + 71*x^5 + 461*x^6 + 3447*x^7 + 29093*x^8 + ...
From _Peter Luschny_, Aug 03 2022: (Start)
A permutation p in [n] (where n >= 0) is reducible if there exists an i in 1..n-1 such that for all j in the range 1..i and all k in the range i+1..n it is true that p(j) < p(k). (Note that a range a..b includes a and b.) If such an i exists we say that i splits the permutation at i.
Examples:
* () is not reducible since there is no index i which splits (). (=> a(0) = 1)
* (1) is not reducible since there is no index i which splits (1). (=> a(1) = 1)
* (1, 2) is reducible since index 1 splits (1, 2) as p(1) < p(2).
* (2, 1) is not reducible since at the only potential splitting point i = 1 we have p(1) > p(2). (=> a(2) = 1)
* For n = 3 we have (1, 2, 3), (1, 3, 2), and (2, 1, 3) are reducible and (2, 3, 1), (3, 1, 2), and (3, 2, 1) are irreducible. (End)

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 25, Example 20.
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M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 22.
H. P. Robinson, Letter to N. J. A. Sloane, Nov 19 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 1, Ex. 128; Vol. 2, 1999, see Problem 5.13(b).

Seiichi Manyama, Table of n, a(n) for n = 0..449 (first 102 terms from T. D. Noe)
Marcelo Aguiar and Frank Sottile, Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, arXiv:math/0203282 [math.CO], 2002-2005.
Marcelo Aguiar and Aaron Lauve, Convolution Powers of the Identity, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. Discrete Mathematics and Theoretical Computer Science, pp.1053-1064, 2013, DMTCS Proceedings. .
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Joerg Arndt, Generating Random Permutations, PhD thesis, Australian National University, Canberra, Australia, (2010).
Joerg Arndt, Matters Computational (The Fxtbook), p. 281
Roland Bacher and Christophe Reutenauer, Number of right ideals and a q-analogue of indecomposable permutations, arXiv preprint arXiv:1511.00426 [math.CO], 2015.
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E. W. Bowen and N. J. A. Sloane, Correspondence, 1976.
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David Callan, Counting Stabilized-Interval-Free Permutations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.8.
Peter Cameron's Blog, The symmetric group, 11, Posted 09/04/2011.
Mahir Bilen Can, Luke Nelson, and Kevin Treat, A Catalanization Map on the Symmetric Group, Enum. Comb. Appl. (2022) Vol. 2, No. 4, #S4PP7.
Daniel Chen and Sebastian Ohlig, Associated Permutations of Complete Non-Ambiguous Trees and Zubieta's Conjecture, arXiv:2210.11117 [math.CO], 2022.
Fan Chung and R. L. Graham, Primitive juggling sequences, Am. Math. Monthly 115 (3) (2008) 185-194.
L. Comtet, Sur les coefficients de l'inverse de la série formelle Sum n! t^n, Comptes Rend. Acad. Sci. Paris, A 275 (1972), 569-572.
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Xinle Dai, Jordan Long, and Karen Yeats, Subdivergence-free gluings of trees, arXiv:2106.07494 [math.CO], 2021.
M. A. Deryagina and A. D. Mednykh, On the enumeration of circular maps with given number of edges, Siberian Mathematical Journal, 54, No. 6, 2013, 624-639.
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J. D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969) 199-205.
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See A167894 for another version.
Bisections give A272656, A272657.
Row sums of A111184 and A089949.
Leading diagonal of A059438. A diagonal of A263484.
Cf. A051296, A084938, A074664, A113869, A144107.
Cf. A260503, A259472, A261214, A261239, A261253, A261254.
Cf. A090238, A000698, A356291 (reducible permutations).
Column k=0 of A370380 and A370381 (without pair of initial terms and with different offset).

INVERTi([seq(n!,n=1..20)]);
A003319 := proc(n) option remember; n! - add((n-j)!*A003319(j), j=1..n-1) end;
[seq(A003319(n), n=0..50)]; # N. J. A. Sloane, Dec 28 2011
series(2 - 1/hypergeom([1,1], [], x), x=0,50); # Mark van Hoeij, Apr 18 2013

a[n_] := a[n] = n! - Sum[k!*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 11 2011, after given formula *)
CoefficientList[Assuming[Element[x,Reals],Series[2-E^(1/x)* x/ExpIntegralEi[1/x],{x,0,20}]],x] (* Vaclav Kotesovec, Mar 07 2014 *)
a[ n_] := If[ n < 2, 1, a[n] = (n - 2) a[n - 1] + Sum[ a[k] a[n - k], {k, n - 1}]]; (* Michael Somos, Feb 23 2015 *)
Table[SeriesCoefficient[1 + x/(1 + ContinuedFractionK[-Floor[(k + 2)/2]*x, 1, {k, 1, n}]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 29 2017 *)

{a(n) = my(A); if( n<1, 1, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (k - 2) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 24 2011 */

{if(n<1,1,a(n)=local(A=x);for(i=1,n,A=x-x*A+A^2+x^2*A' +x*O(x^n));polcoeff(A,n))} /* Paul D. Hanna, Jul 30 2011 */

def A003319_list(len):
    R, C = [1], [1] + [0] * (len - 1)
    for n in range(1, len):
        for k in range(n, 0, -1):
            C[k] = C[k - 1] * k
        C[0] = -sum(C[k] for k in range(1, n + 1))
        R.append(-C[0])
    return R
print(A003319_list(21))  # Peter Luschny, Feb 19 2016

G.f.: 2 - 1/Sum_{k>=0} k!*x^k.
Also a(n) = n! - Sum_{k=1..n-1} k!*a(n-k) [Bowen, 1976].
Also coefficients in the divergent series expansion log Sum_{n>=0} n!*x^n = Sum_{n>=1} a(n+1)*x^n/n [Bowen, 1976].
a(n) = (-1)^(n-1) * det {| 1! 2! ... n! | 1 1! ... (n-1)! | 0 1 1! ... (n-2)! | ... | 0 ... 0 1 1! |}.
INVERTi transform of factorial numbers, A000142 starting from n=1. - Antti Karttunen, May 30 2003
Gives the row sums of the triangle [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938; this triangle A089949. - Philippe Deléham, Dec 30 2003
a(n+1) = Sum_{k=0..n} A089949(n,k). - Philippe Deléham, Oct 16 2006
L.g.f.: Sum_{n>=1} a(n)*x^n/n = log( Sum_{n>=0} n!*x^n ). - Paul D. Hanna, Sep 19 2007
G.f.: 1+x/(1-x/(1-2*x/(1-2*x/(1-3*x/(1-3*x/(1-4*x/(1-4*x/(1-...)))))))) (continued fraction). - Paul Barry, Oct 07 2008
a(n) = -Sum_{i=0..n} (-1)^i*A090238(n, i) for n > 0. - Peter Luschny, Mar 13 2009
From Gary W. Adamson, Jul 14 2011: (Start)
a(n) = upper left term in M^(n-1), M = triangle A128175 as an infinite square production matrix (deleting the first "1"); as follows:
   1,  1,  0,  0,  0,  0, ...
   2,  2,  1,  0,  0,  0, ...
   4,  4,  3,  1,  0,  0, ...
   8,  8,  7,  4,  1,  0, ...
  16, 16, 15, 11,  5,  1, ...
  ... (End)
O.g.f. satisfies: A(x) = x - x*A(x) + A(x)^2 + x^2*A'(x). - Paul D. Hanna, Jul 30 2011
From Sergei N. Gladkovskii, Jun 24 2012: (Start)
Let A(x) be the g.f.; then
A(x) = 1/Q(0), where Q(k) = x + 1 + x*k - (k+2)*x/Q(k+1).
A(x) = (1-1/U(0))/x, when U(k) = 1 + x*(2*k+1)/(1 - 2*x*(k+1)/(2*x*(k+1) + 1/U(k+1))). (End)
From Sergei N. Gladkovskii, Aug 03 2013: (Start)
Continued fractions:
G.f.: 1 - G(0)/2, where G(k) = 1 + 1/(1 - x*(2*k+2)/(x*(2*k+2) - 1 + x*(2*k+2)/G(k+1))).
G.f.: (x/2)*G(0), where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1/2) + 1/G(k+1))).
G.f.: x*G(0), where G(k) = 1 - x*(k+1)/(x - 1/G(k+1)).
G.f.: 1 - 1/G(0), where G(k) = 1 - x*(k+1)/(x*(k+1) - 1/(1 - x*(k+1)/(x*(k+1) - 1/G(k+1)))).
G.f.: x*W(0), where W(k) = 1 - x*(k+1)/(x*(k+1) - 1/(1 - x*(k+2)/(x*(k+2) - 1/W(k+1)))).
(End)
a(n) = A233824(n-1) if n > 0. (Proof. Set b(n) = A233824(n), so that b(n) = n*n! - Sum_{k=1..n-1} k!*b(n-k). To get a(n+1) = b(n) for n >= 0, induct on n, use (n+1)! = n*n! + n!, and replace k with k+1 in the sum.) - Jonathan Sondow, Dec 19 2013
a(n) ~ n! * (1 - 2/n - 1/n^2 - 5/n^3 - 32/n^4 - 253/n^5 - 2381/n^6 - 25912/n^7 - 319339/n^8 - 4388949/n^9 - 66495386/n^10), for coefficients see A260503. - Vaclav Kotesovec, Jul 27 2015
For n>0, a(n) = (A059439(n) - A259472(n))/2. - Vaclav Kotesovec, Aug 03 2015
From Peter Bala, May 23 2017: (Start)
G.f.: 1 + x/(1 + x - 2*x/(1 + 2*x - 3*x/(1 + 3*x - 4*x/(1 + 4*x - ...)))). Cf. A000698.
G.f.: 1/(1 - x/(1 + x - x/(1 - 2*x/(1 - 2*x/(1 - 3*x/(1 - 3*x/(1 - 4*x/(1 - 4*x/(1 - ...))))))))). (End)
Conjecture: a(n) = A370380(n-2, 0) = A370381(n-2, 0) for n > 1 with a(0) = a(1) = 1. - Mikhail Kurkov, Apr 26 2024

More terms from Michael Somos, Jan 26 2000
Additional comments from Marcelo Aguiar (maguiar(AT)math.tamu.edu), Mar 28 2002
Added a(0)=0 (some of the formulas may now need adjusting). - N. J. A. Sloane, Sep 12 2012
Edited and set a(0) = 1 by Peter Luschny, Aug 03 2022

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
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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

A112354 Inverse Euler transform of n!. Also the number of sequences of permutations with no global descents which are Lyndon (smallest in lexicographic order of all cyclic shifts of the sequences) where the size of the sequence = sum of sizes of the permutations.

Original entry on oeis.org

1, 1, 4, 17, 92, 572, 4156, 34159, 314368, 3199844, 35703996, 433421495, 5687955724, 80256874912, 1211781887796, 19496946534720, 333041104402860, 6019770246910128, 114794574818830716, 2303332661416242633, 48509766592884311132, 1069983257387132347080

Offset: 1

Mike Zabrocki, Sep 05 2005

nonn

			a(3) = 4 because (123), (213), (132) and (1,21) are all Lyndon.
a(4) = 17 because there are 13 permutations with no global descents of size 4 and (1,123), (1,213), (1,132) are all Lyndon.
a(5) = 92 = 71 permutations with no global descents+13 sequences of the form (1,pi) where pi in S_4 with no global descents+(1,1,1,21),(1,21,21),(1,1,123),(1,1,213),(1,1,132),(21,123),(21,213),(21,132).

Seiichi Manyama, Table of n, a(n) for n = 1..449 (terms 1..200 from Alois P. Heinz)
M. Aguiar and A. Lauve, Antipode and Convolution Powers of the Identity in Graded Connected Hopf Algebras, FPSAC 2013 Paris, France DMTCS Proc. AS, 2013, 1083-1094.
M. Aguiar, A. Lauve, The characteristic polynomial of the Adams operators on graded connected Hopf algebras, arXiv:1403.7584 [math.RA], 2014.

Cf. A003319, A000142, A305786.

read transforms; EULERi([seq(n!,n=1..30)]);
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(factorial):
seq(a(n), n = 1..22); # Peter Luschny, Nov 21 2022

ff = Range[n = 22]!; s = {}; For[i = 1, i <= n, i++, AppendTo[s, i*ff[[i]] - Sum[s[[d]]*ff[[i-d]], {d, i-1}]]]; Table[Sum[If[Divisible[i, d], MoebiusMu[i/d], 0]*s[[d]], {d, 1, i}]/i, {i, n}] (* Jean-François Alcover, Apr 15 2016 *)

Product_{k>=1} 1/(1-x^k)^{a(k)} = Sum_{n>=0} n! x^n.

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 coefficients see A113869. - Vaclav Kotesovec, Sep 04 2014, extended Nov 27 2020

A168246 Inverse Weigh transform of n!.

Original entry on oeis.org

1, 2, 4, 19, 92, 576, 4156, 34178, 314368, 3199936, 35703996, 433422071, 5687955724, 80256879068, 1211781887796, 19496946568898, 333041104402860, 6019770247224496, 114794574818830716, 2303332661419442569, 48509766592884311132, 1069983257387168051076

Offset: 1

Vladeta Jovovic, Nov 21 2009

Alois P. Heinz, Table of n, a(n) for n = 1..449

Cf. A000142, A112354, A261052 (Weigh transform of n!).

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; n! -b(n, n-1) end:
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 11 2018

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := a[n] = n! - b[n, n - 1];
Array[a, 30] (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz *)

seq(n)={dirdiv(Vec(log(1+x*Ser(vector(n, n, n!)))), -vector(n, n, (-1)^n/n))} \\ Andrew Howroyd, Jun 22 2018

Product_{k>=1} (1+x^k)^a(k) = Sum_{n>=0} n! x^n.

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 coefficients see A113869. - Vaclav Kotesovec, Nov 27 2020

A113871 G.f.: 1/(Sum_{k>=0} (k!)^2 x^k).

Original entry on oeis.org

1, -1, -3, -29, -499, -13101, -486131, -24266797, -1571357619, -128264296301, -12894743113075, -1566235727656365, -226180775756251955, -38308065207361046509, -7521255169156107737331, -1694604321825062440852013, -434302821056087233474158259

Offset: 0

N. J. A. Sloane, Jan 26 2006

sign

T. D. Noe, Table of n, a(n) for n = 0..100
Marcelo Aguiar and Swapneel Mahajan, On The Hadamard product Of Hopf monoids
John D. Dixon, Asymptotics of Generating the Symmetric and Alternating Groups, Electronic Journal of Combinatorics, 2005, vol 11(2), R56.

Cf. A003319, A051296, A113869, A114038, A316862.

nn = 20; CoefficientList[Series[1/Sum[(k!)^2 x^k, {k, 0, nn}], {x, 0, nn}], x] (* T. D. Noe, Jan 03 2013 *)

h = 1/(1+x*hypergeometric((1,2,2),(),x))
taylor(h,x,0,16).list() # Peter Luschny, Jul 28 2015

def A113871_list(len):
    R, C = [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        for k in range(n,-1,-1):
            C[k] = C[k-1] * k^2
        C[0] = -sum(C[k] for k in (1..n))
        R.append(C[0])
    return R
print(A113871_list(17)) # Peter Luschny, Jul 30 2015

G.f.: 2/Q(0), where Q(k) = 1 + 1/(1 - (k+1)^2*x/((k+1)^2*x + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 17 2013
a(n) ~ -n!^2 * (1 - 2/n^2 - 5/n^4 - 10/n^5 - 67/n^6 - 332/n^7 - 2152/n^8 - 14946/n^9 - 115583/n^10). - Vaclav Kotesovec, Jul 28 2015
a(0) = 1, a(n) = -Sum_{k=0..n-1} a(k) * ((n-k)!)^2. - Daniel Suteu, Feb 23 2018

A122949 Number of ordered pairs of permutations generating a transitive group.

Original entry on oeis.org

1, 3, 26, 426, 11064, 413640, 20946960, 1377648720, 114078384000, 11611761920640, 1425189271161600, 207609729886944000, 35419018603306060800, 6996657393055480550400, 1584616114318716544665600, 407930516160959891683584000, 118458533875304716189544448000

Offset: 1

Philippe Flajolet, Oct 25 2006

nonn

From Dixon: The sequence is asymptotic to (n!)^2; when divided by n!^2, it has a high-order asymptotic contact with the probability that two randomly chosen permutations generate the symmetric group. Also: a(n)=(n-1)!*A003319(n+1), where A003319 is the number of connected [or indecomposable] permutations. The coefficients in the asymptotic expansion of a(n)/(n!)^2 are A113869 and in absolute value, they constitute A084357 (number of sets of sets of lists).

			a(2)=3 because there are 2!*2!=4 pairs of permutations, of which only [(1,1),(1,1)] does not generate a transitive group.

Seiichi Manyama, Table of n, a(n) for n = 1..253
John D. Dixon, Asymptotics of Generating the Symmetric and Alternating Groups, Electronic Journal of Combinatorics, vol 11(2), R56.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; page 139.

Cf. A003319, A084357, A113869.

series(log(add(n!*z^n,n=0..Order+2)),z=0):seq(coeff(%,z,j)*j!,j=0..Order);

max = 15; Drop[ CoefficientList[ Series[ Log[1 + Sum[n!*z^n, {n, 1, max}]], {z, 0, max}], z]* Range[0, max]!, 1](* Jean-François Alcover, Oct 05 2011 *)

N=20; x='x+O('x^N); Vec(serlaplace(log(sum(k=0, N, k!*x^k)))) \\ Seiichi Manyama, Mar 01 2019

Exponential generating function is: log(1+Sum_{n>=1}n!*z^n).

a(n) = (n!)^2 - (n-1)! * Sum_{k=1..n-1} a(k) * (n-k)! / (k-1)!. - Ilya Gutkovskiy, Jul 10 2020

More terms from Seiichi Manyama, Mar 01 2019

A158094 G.f. Product_{n>=1} (1 + a(n)x^n) = Sum_{n>=0} n!x^n.

Original entry on oeis.org

1, 2, 4, 20, 92, 580, 4156, 34372, 314348, 3204116, 35703996, 433587396, 5687955724, 80265513140, 1211781628060, 19497537309028, 333041104402860, 6019819589363348, 114794574818830716, 2303337794614783236

Offset: 1

Paul D. Hanna, Apr 15 2009

nonn

Robert Israel, Table of n, a(n) for n = 1..450

Cf. A316084.

A158094:= proc(n)
option remember;
local S;
S:= series(add(k!*x^k,k=0..n)/mul(1+A158094(k)*x^k,k=1..n-1),x,n+1);
coeff(S,x,n)
end; # Robert Israel, Mar 04 2014

a[n_] := a[n] = Module[{s}, s = Series[Sum[k!*x^k, {k, 0, n}]/Product[1+a[k]*x^k, {k, 1, n-1}], {x, 0, n+1}]; Coefficient[s, x, n]]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Mar 04 2014, after Maple *)

{a(n)=if(n<1, 0, polcoeff(sum(k=0,n,k!*x^k)/prod(k=1, n-1, 1+a(k)*x^k +x*O(x^n)), n))}

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 coefficients see A113869. - Vaclav Kotesovec, Jun 18 2019

A316084 Product_{k>=1} 1/(1 - a(k)x^k) = Sum_{k>=0} k!x^k.

Original entry on oeis.org

1, 1, 4, 17, 92, 566, 4156, 34023, 314348, 3195658, 35703996, 433259908, 5687955724, 80248240822, 1211781628060, 19496367748659, 333041104402860, 6019720779293770, 114794574818830716, 2303327555284622304, 48509766568956367372, 1069982619999485015070

Offset: 1

Seiichi Manyama, Jun 23 2018

nonn

			1/((1-x)*(1-x^2)*(1-4*x^3)*(1-17*x^4)* ... ) = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + ... .

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

Cf. A000142, A113869, A158094.

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 coefficients see A113869. - Vaclav Kotesovec, Jun 18 2019

a(2*n-1) = A158094(2*n-1). - Vaclav Kotesovec, Jun 18 2019

A306153 Inverse Weigh transform of (-1)^n * n!.

Original entry on oeis.org

-1, 1, -4, 18, -92, 572, -4156, 34177, -314368, 3199844, -35703996, 433422067, -5687955724, 80256874912, -1211781887796, 19496946568897, -333041104402860, 6019770246910128, -114794574818830716, 2303332661419442477, -48509766592884311132, 1069983257387132347080

Offset: 1

Seiichi Manyama, Jun 23 2018

sign

			(1+x)^(-1)*(1+x^2)*(1+x^3)^(-4)*(1+x^4)^18* ... = 1 - x + 2*x^2 - 6*x^3 + 24*x^4 - ... .

Alois P. Heinz, Table of n, a(n) for n = 1..449
N. J. A. Sloane, Transforms

Cf. A168246.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> (-1)^n*n!-b(n, n-1):
seq(a(n), n=1..24);  # Alois P. Heinz, Jun 23 2018

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]];
a[n_] := (-1)^n n! - b[n, n - 1] // FullSimplify;
Array[a, 24] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

a(n) ~ (-1)^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 coefficients see A113869. - Vaclav Kotesovec, Nov 27 2020

cp's OEIS Frontend

A114038 Analog of A113869 for three generators.

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A003319 Number of connected permutations of [1..n] (those not fixing [1..j] for 0 < j < n). Also called indecomposable permutations, or irreducible permutations.

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

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A112354 Inverse Euler transform of n!. Also the number of sequences of permutations with no global descents which are Lyndon (smallest in lexicographic order of all cyclic shifts of the sequences) where the size of the sequence = sum of sizes of the permutations.

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A168246 Inverse Weigh transform of n!.

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A113871 G.f.: 1/(Sum_{k>=0} (k!)^2 x^k).

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A122949 Number of ordered pairs of permutations generating a transitive group.

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A158094 G.f. Product_{n>=1} (1 + a(n)x^n) = Sum_{n>=0} n!x^n.

A316084 Product_{k>=1} 1/(1 - a(k)x^k) = Sum_{k>=0} k!x^k.