A003319 -id:A003319 - cp's OEIS frontend

A065164 Permutation t->t+1 of Z, folded to N.

2, 4, 1, 6, 3, 8, 5, 10, 7, 12, 9, 14, 11, 16, 13, 18, 15, 20, 17, 22, 19, 24, 21, 26, 23, 28, 25, 30, 27, 32, 29, 34, 31, 36, 33, 38, 35, 40, 37, 42, 39, 44, 41, 46, 43, 48, 45, 50, 47, 52, 49, 54, 51, 56, 53, 58, 55, 60, 57, 62, 59, 64, 61, 66, 63, 68, 65, 70, 67, 72, 69, 74

Offset: 1

Antti Karttunen, Oct 19 2001

Corresponds to simple periodic asynchronic site swap pattern ...111111... (tossing one ball from hand to hand forever).
This permutation consists of a single infinite cycle.
This is, starting at a(2) = 4, the same as the "increasing oscillating sequence" shown in Proposition 3.1, p.7 and plotted in the right of figure 1, of Vatter. The same paper, p.4, cites Comtet and uses without giving the A-number of A003319. Abstract: We prove that there are permutation classes (hereditary properties of permutations) of every growth rate (Stanley-Wilf limit) at least lambda = approx 2.48187, the unique real root of x^5-2x^4-2x^2-2x-1, thereby establishing a conjecture of Albert and Linton. - Jonathan Vos Post, Jul 18 2008

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 819.

Michael H. Albert, Robert Brignall, and Vincent Vatter, Large infinite antichains of permutations, arXiv:1212.3346 [math.CO], 2012.
Joe Buhler and R. L. Graham, Juggling Drops and Descents, Amer. Math. Monthly, 101, (no. 6) 1994, 507-519.
Jay Pantone and Vincent Vatter, Growth rates of permutation classes: categorization up to the uncountability threshold, arXiv:1605.04289 [math.CO], 2016-2019.
Vincent Vatter, Permutation classes of every growth rate above 2.48188, arXiv:0807.2815 [math.CO], 2008-2009.
Vincent Vatter, Permutation classes, arXiv:1409.5159 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
Index entries for sequences that are permutations of the natural numbers.

Row 1 of A065167. Obtained by composing permutations A014681 and A065190. Inverse permutation: A065168.

ss1 := [seq(PerSS(n,1), n=1..120)]; PerSS := (n,c) -> Z2N(N2Z(n)+c);
N2Z := n -> ((-1)^n)*floor(n/2); Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);

Join[{2}, LinearRecurrence[{1, 1, -1}, {4, 1, 6}, 100]] (* Amiram Eldar, Aug 08 2023 *)

Let f: Z -> N be given by f(z) = 2z if z>0 else 2|z|+1, with inverse g(z) = z/2 if z even else (1-z)/2. Then a(n) = f(g(n)+1).
a(n) = n + 2*(-1^n) for n > 1. - Frank Ellermann, Feb 12 2002
a(n) = 2*n-a(n-1)-1, n>2. - Vincenzo Librandi, Dec 07 2010, corrected by R. J. Mathar, Dec 07 2010
From Colin Barker, Feb 18 2013: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n>4.
G.f.: x*(3*x^3-5*x^2+2*x+2) / ((x-1)^2*(x+1)). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) + 1. - Amiram Eldar, Aug 08 2023

A109062 Triangle read by rows: number of atomic set compositions of size n and length k (see description in A095989) 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 11, 23, 13, 1, 26, 112, 158, 71, 1, 57, 446, 1170, 1241, 461, 1, 120, 1593, 6880, 12871, 10912, 3447, 1, 247, 5337, 35503, 103887, 150413, 106031, 29093, 1, 502, 17190, 168982, 724148, 1589266, 1872286, 1128218, 273343, 1, 1013, 54008

Offset: 1

Mike Zabrocki, Aug 24 2005

Also the number of free generators and primitives of the quasi-symmetric functions in non-commuting variables. - Mike Zabrocki, Aug 06 2006
Triangle given by [1,0,2,0,3,0,4,0,5,...] DELTA [1,2,2,3,3,4,4,5,5,6,6,7,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 01 2007
Apparently, the alternating sums vanish for n > 1. - F. Chapoton, Sep 05 2023

			Atomic set compositions a(1,1)=1: [{1}]; a(2,1)=1, a(2,2)=1: [{12}], [{2},{1}]; a(3,1)=1, a(3,2)=4, a(3,3)=3: [{123}], [{2},{13}], [{3}, {12}], [{23}, {1}], [{13},{2}], [{2},{3},{1}], [{3},{1},{2}], [{3},{2},{1}].
Triangle begins:
  1;
  1,  1;
  1,  4,   3;
  1, 11,  23,  13;
  1, 26, 112, 158, 71;
  ...

N. Bergeron and M. Zabrocki, The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree, arXiv:math/0509265 [math.CO], 2005.
Ming-Jian Ding and Bao-Xuan Zhu, Some results related to Hurwitz stability of combinatorial polynomials, Advances in Applied Mathematics, Volume 152, (2024), 102591. See p. 7.

Row sums are equal to A095989, a(n,n) = A003319, a(n,2) = A000295.

Cf. A095989, A059438, A074664, A087903, A008277, A019538.

f:=(n,k)->coeff(coeff(series(1-1/(1+add(add(q^m*t^i*
    Stirling2(m,i)*i!,i=1..m),m=1..n)),q,n+1),q,n),t,k):
seq(seq(f(n,k), k=1..n), n=1..10);

G.f.: 1-1/(1+Sum_{n>=1} Sum_{k=1..n} q^n*t^k*Stirling2(n,k)*k!).

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

A140456 a(n) is the number of indecomposable involutions of length n.

Original entry on oeis.org

1, 1, 1, 3, 7, 23, 71, 255, 911, 3535, 13903, 57663, 243871, 1072031, 4812575, 22278399, 105300287, 510764095, 2527547455, 12794891007, 66012404863, 347599231103, 1863520447103, 10178746224639, 56548686860543, 319628408814847, 1835814213846271

Offset: 1

Joel B. Lewis, Jul 22 2008

nonn

An involution is a self-inverse permutation. A permutation of [n] = {1, 2, ..., n} is indecomposable if it does not fix [j] for any 0 < j < n.
From Paul Barry, Nov 26 2009: (Start)
G.f. of a(n+1) is 1/(1-x-2x^2/(1-x-3x^2/(1-x-4x^2/(1-x-5x^2/(1-...))))) (continued fraction).
a(n+1) is the binomial transform of the aeration of A000698(n+1). Hankel transform of a(n+1) is A000178(n+1). (End)
From Groux Roland, Mar 17 2011: (Start)
a(n) is the INVERTi transform of A000085(n+1)
a(n) is also the moment of order n for the density: sqrt(2/Pi^3)*exp((x-1)^2/2)/(1-(erf(I*(x-1)/sqrt(2)))^2).
More generally, if c(n)=int(x^n*rho(x),x=a..b) with rho(x) a probability density function of class C1, then the INVERTi transform of (c(1),..c(n),..) starting at n=2 gives the moments of mu(x) = rho(x) / ((s(x))^2+(Pi*rho(x))^2) with s(x) = int( rho'(t)*log(abs(1-t/x)), t=a..b) + rho(b)*log(x/(b-x)) + rho(a)*log((x-a)/x).
(End)
For n>1 sum over all Motzkin paths of length n-2 of products over all peaks p of (x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p. - Alois P. Heinz, May 24 2015

			The unique indecomposable involution of length 3 is 321. The indecomposable involutions of length 4 are 3412, 4231 and 4321.
G.f. = x + x^2 + 3*x^3 + 7*x^4 + 23*x^5 + 71*x^6 + 255*x^7 + 911*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 1..800 (terms n = 1..50 from Joel B. Lewis)
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Claudia Malvenuto and Christophe Reutenauer, Primitive Elements of the Hopf Algebras of Tableaux, arXiv:2010.06731 [math.CO], 2020.

Cf. A000085 (involutions), A000698 (indecomposable fixed-point free involutions), and A003319 (indecomposable permutations).

Cf. A001006, A258306.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, (x+y)/y, 1)
                 + b(x-1, y, false) + b(x-1, y+1, true)))
    end:
a:= n-> `if`(n=1, 1, b(n-2, 0, false)):
seq(a(n), n=1..35);  # Alois P. Heinz, May 24 2015

CoefficientList[Series[1 - 1/Total[CoefficientList[Series[E^(x + x^2/2), {x, 0, 50}], x] * Range[0, 50]! * x^Range[0, 50]], {x, 0, 50}], x]

G.f.: 1 - 1/I(x), where I(x) is the ordinary generating function for involutions (A000085).

G.f.: Q(0) +1/x, where Q(k) = 1 - 1/x - (k+1)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Sep 16 2013

A167894 Expansion of g.f.: 1/(Sum_{k >= 0} k!*x^k).

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

Offset: 0

Philippe Deléham, Nov 15 2009

sign

Essentially the same as A003319, which is the main entry for these numbers. - N. J. A. Sloane, Jun 11 2013

M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 40.

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A003319, A158882.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(&+[Factorial(k)*x^k: k in [0..m+1]]) )); // G. C. Greubel, Feb 07 2019

CoefficientList[Series[1/(Sum[k!*x^k, {k, 0, 25}]), {x, 0, 20}], x] (* G. C. Greubel, Jun 30 2016 *)

a(n) := if n=0 then 1 else -sum( a(i)*(n-i)!,i,0,n-1); /* Vladimir Kruchinin, Oct 10 2024 */

m=20; my(x='x+O('x^m)); Vec(1/sum(k=0,m+1, k!*x^k)) \\ G. C. Greubel, Feb 07 2019

def A167894_list(len):
    R, C = [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        for k in range(n, 0, -1):
            C[k] = C[k-1] * k
        C[0] = -sum(C[k] for k in (1..n))
        R.append(C[0])
    return R
print(A167894_list(20)) # Peter Luschny, Feb 19 2016

m=20; (1/sum(factorial(k)*x^k for k in range(m+1))).series(x, m).coefficients(x, sparse=False) # G. C. Greubel, Feb 07 2019

a(n) = - Sum_{i=0..n-1} a(i)*(n-i)! for n > 0 with a(0) = 1. - Vladimir Kruchinin, Oct 10 2024
From Sergei N. Gladkovskii, Jun 24 2012, Oct 15 2012, Nov 18 2012, Dec 26 2012, Apr 25 2013, May 29 2013, Aug 08 2013, Nov 19 2013: (Start) Continued fractions:
G.f.: 1 - x/Q(0), where Q(k) = 1 - (k+1)*x/(1 - (k+2)*x/Q(k+1)).
G.f.: U(0) where U(k) =  1 - x*(k+1)/(1 - x*(k+1)/U(k+1)).
G.f.: 1/G(0) where G(k) = 1 + x*(2*k+1)/(1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))).
G.f.: A(x) = 1 - x/G(0) where G(k) = 1 + (k+1)*x - x*(k+2)/G(k+1).
G.f.: x*Q(0), where Q(k) = 1/x - 1 - 2*k - (k+1)^2/Q(k+1).
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))).
G.f.: 2/Q(0), where Q(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/Q(k+1) )).
G.f.: conjecture: Q(0), where Q(k) = 1 + k*x - (k+1)*x/Q(k+1). (End)
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 - ...). - Vaclav Kotesovec, Dec 08 2020

A233824 A recurrent sequence in Panaitopol's formula for pi(x), where pi(x) is the number of primes <= x.

Original entry on oeis.org

0, 1, 3, 13, 71, 461, 3447, 29093, 273343, 2829325, 31998903, 392743957, 5201061455, 73943424413, 1123596277863, 18176728317413, 311951144828863, 5661698774848621, 108355864447215063, 2181096921557783605

Offset: 0

Jonathan Sondow, Dec 17 2013

nonn

Sum_{k=0..n} k!*a(n-k) = n*n!.

Panaitopol proved that x/pi(x) = log(x) - 1 - Sum_{k=1..m} a(k)/log(x)^k + O(1/log(x)^{m+1}) for m > 0.

			0!*a(0) = a(0) = 0*0!, so a(0) = 0.
0!*a(1) + 1!*a(0) = a(1) + a(0) = 1*1!, so a(1) = 1.
0!*a(2) + 1!*a(1) + 2!*a(0) = a(2) + a(1) + 2*a(0) = 2*2!, so a(2) = 4 - 1 = 3.

G. C. Greubel, Table of n, a(n) for n = 0..445
Safia Aoudjit and Djamel Berkane, Explicit Estimates Involving the Primorial Integers and Applications, J. Int. Seq., Vol. 24 (2021), Article 21.7.8.
M. Hassani, Some remarks on a determinant related to the prime counting function, The 8th Seminar on Linear Algebra and its Applications, 13-14th May 2015, University of Kurdistan, Iran.
A. Ivić, Review of "A formula for pi(x) applied to a result of Koninck-Ivić" by L. Panaitopol, Zbl 0982.11003.
A. Kourbatov, Upper Bounds for Prime Gaps Related to Firoozbakht's Conjecture, J. Integer Sequences, 18 (2015), Article 15.11.2. arXiv:1506.03042
A. Kourbatov, On the geometric mean of the first n primes, arXiv:1603.00855 [math.NT], 2016.
L. Panaitopol, A formula for pi(x) applied to a result of Koninck-Ivić, Nieuw Arch. Wiskd., (5) 1, No. 1 (2000), 55-56.

Cf. A000720, A003319, A062049.

a[0] = 0; a[n_] := a[n] = n*n! - Sum[ k! a[n - k], {k, n - 1}]; Table[a@ n, {n, 0, 19}] (* Michael De Vlieger, Mar 26 2016 *)

a(n) = n*n! - Sum_{k=1..n-1} k!*a(n-k).

a(n) = A003319(n+1) if n > 0. (Proof. Set b(n) = A003319(n), so that b(n) = n! - Sum_{k=1..n-1} k!*b(n-k). To get b(n+1) = a(n) for n > 0, induct on n, use (n+1)! = n*n! + n!, and replace k with k+1 in the sum.)

A004208 a(n) = n * (2n - 1)!! - Sum_{k=0..n-1} a(k) (2n - 2k - 1)!!.

Original entry on oeis.org

1, 5, 37, 353, 4081, 55205, 854197, 14876033, 288018721, 6138913925, 142882295557, 3606682364513, 98158402127761, 2865624738913445, 89338394736560917, 2962542872271918593, 104128401379446177601, 3867079042971339087365, 151312533647578564021477

Offset: 1

N. J. A. Sloane, following a suggestion from E. W. Bowen, Aug 27 1976

nonn

a(n+1) is the moment of order n for the probability density function rho(x) = Pi^(-3/2)*sqrt(x/2)*exp(x/2)/(1-erf^2(i*sqrt(x/2))) on the interval 0..infinity, where erf is the error function and i=sqrt(-1). - Groux Roland, Nov 10 2009

E. W. Bowen, Letter to N. J. A. Sloane, Aug 27 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..300
E. W. Bowen & N. J. A. Sloane, Correspondence, 1976
Colin K, Is it possible to make Mathematica reformulate an expression in a more numerically stable way?. see second answer by whuber.

Cf. A000698.

df := proc(n) product(2*k-1,k=1..n) end: a[1] := 1: for n from 2 to 30 do a[n] := n*df(n)-sum(a[k]*df(n-k),k=1..n-1) od;

CoefficientList[Series[D[Log[Sum[(2n-1)!!x^n,{n,0,19}]],x],{x,0,18}],x] (* Wouter Meeussen, Mar 21 2009 *)
a[ n_] := If[ n < 1, 0, n Coefficient[ Normal[ Series[ Log @ Erfc @ Sqrt @ x, {x, Infinity, n}] + x + Log[ Sqrt [Pi x]]] /. x -> -1 / 2 / x, x, n]] (* Michael Somos, May 28 2012 *)

{a(n) = if( n<1, 0, n++; polcoeff( 1 - 1 / (2 * sum( k=0, n, x^k * (2*k)! / (2^k * k!), x * O(x^n))), n))} /* Michael Somos, May 28 2012 */

a(n) = (1/2) * A000698(n+1), n > 0.
x + (5/2)*x^2 + (37/3)*x^3 + (353/4)*x^4 + (4081/5)*x^5 + (55205/6)*x^6 + ... = log(1 + x + 3*x^2 + 15*x^3 + 105*x^4 + 945*x^5 + 10395*x^6 + ...) where [1, 1, 3, 15, 105, 945, 10395, ...] = A001147(double factorials). - Philippe Deléham, Jun 20 2006
G.f.: ( 1/Q(0) - 1)/x where Q(k) = 1 - x*(2*k+1)/(1 - x*(2*k+4)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013
G.f.: (2/x)/G(0) - 1/x, where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) - 1 + 2*x*(2*k+4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013
G.f.: 1/(2*x^2) - 1/(2*x) - G(0)/(2*x^2), where G(k) = 1 - x*(k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Aug 15 2013
L.g.f.: log(1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - ...))))))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 10 2017

Description corrected by Jeremy Magland (magland(AT)math.byu.edu), Jan 07 2000

More terms from Emeric Deutsch, Dec 21 2003

A049294 Number of subgroups of index 3 in free group of rank n+1.

Original entry on oeis.org

1, 13, 97, 625, 3841, 23233, 139777, 839425, 5038081, 30231553, 181395457, 1088385025, 6530334721, 39182057473, 235092443137, 1410554855425, 8463329525761, 50779977940993, 304679869218817, 1828079218458625

Offset: 0

Valery A. Liskovets

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
M. Hall, Subgroups of finite index in free groups, Canad. J. Math., 1 (1949), 187-190.
V. A. Liskovets and A. Mednykh, Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.
Index entries for linear recurrences with constant coefficients, signature (9,-20,12).

Cf. A003319, A027837, A049290, A049291, A049292, A049293, A049295.

LinearRecurrence[{9,-20,12},{1,13,97},20] (* Harvey P. Dale, Sep 24 2017 *)

a(n) = 3*6^n-3*2^n+1.

G.f.: (1+4*x)/((1-x)*(1-2*x)*(1-6*x)). [Colin Barker, May 08 2012]

More terms from Karen Richardson (s1149414(AT)cedarville.edu)

A059439 A diagonal of A059438.

Original entry on oeis.org

0, 0, 1, 2, 7, 32, 177, 1142, 8411, 69692, 642581, 6534978, 72754927, 880877928, 11530686953, 162331760494, 2446380427331, 39300220067668, 670480457586813, 12106985274788506, 230691361507912471, 4625811718758963136

Offset: 0

N. J. A. Sloane, Feb 01 2001

Self-convolution of A003319. - Vaclav Kotesovec, Aug 03 2015

			G.f. = x^2 + 2*x^3 + 7*x^4 + 32*x^5 + 177*x^6 + 1142*x^7 + 8411*x^8 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 262 (#14).

Cf. A003319, A259472, A260503, A260913.

a[0]=0; a[n_]:=a[n] = n!-Sum[k!*a[n-k], {k,1,n-1}]; Table[Sum[a[k]*a[n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 03 2015 *)
CoefficientList[Assuming[Element[x, Reals], Series[(1 - x*E^(1/x) / ExpIntegralEi[1/x])^2, {x, 0, 20}]], x] (* Vaclav Kotesovec, Aug 03 2015 *)

G.f.: (1-1/Sum (k! x^k ))^2.
For n>0, a(n) = A259472(n) + 2*A003319(n). - Vaclav Kotesovec, Aug 03 2015
a(n) ~ 2*(n-1)! * (1 - 1/n - 1/n^2 + 1/n^3 + 30/n^4 + 404/n^5 + 5379/n^6 + 76021/n^7 + 1155805/n^8 + 18931873/n^9 + 333434490/n^10), for coefficients see A260913. - Vaclav Kotesovec, Aug 03 2015

More terms from Vladeta Jovovic, Mar 04 2001

Prepended a(0)=0, a(1)=0 from Vaclav Kotesovec, Aug 03 2015

cp's OEIS Frontend

A065164 Permutation t->t+1 of Z, folded to N.

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A109062 Triangle read by rows: number of atomic set compositions of size n and length k (see description in A095989) 1 <= k <= 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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A140456 a(n) is the number of indecomposable involutions of length n.

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A167894 Expansion of g.f.: 1/(Sum_{k >= 0} k!*x^k).

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A233824 A recurrent sequence in Panaitopol's formula for pi(x), where pi(x) is the number of primes <= x.

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A004208 a(n) = n * (2n - 1)!! - Sum_{k=0..n-1} a(k) (2n - 2k - 1)!!.