A114038 -id:A114038 - cp's OEIS frontend

A113869 Coefficients in asymptotic expansion of probability that a random pair of elements from the alternating group A_k generates all of A_k.

1, -1, -1, -4, -23, -171, -1542, -16241, -194973, -2622610, -39027573, -636225591, -11272598680, -215668335091, -4431191311809, -97316894892644, -2275184746472827, -56421527472282127, -1479397224086870294, -40897073524132164189, -1188896226524012279617

Offset: 0

N. J. A. Sloane, Jan 26 2006

Vaclav Kotesovec, Table of n, a(n) for n = 0..420
L. Babai, The probability of generating the symmetric group, J. Combin. Theory, A52 (1989), 148-153.
J. Bovey and A. Williamson, The probability of generating the symmetric group, Bull. London Math. Soc. 10 (1978) 91-96.
J. D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969) 199-205.
J. D. Dixon, Asymptotics of Generating the Symmetric and Alternating Groups, Electronic Journal of Combinatorics, vol 11(2), R56.
Thibault Godin, An analogue to Dixon's theorem for automaton groups, arXiv preprint arXiv:1610.03301 [math.GR], 2016.
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A003319, A112354, A113871, A114038, A158094, A168246, A306153, A316084.

A003319[n_] := A003319[n] = n! - Sum[ k!*A003319[n-k], {k, 1, n-1}]; a[n_] := -Sum[ A003319[i]*StirlingS2[n-1, i-1], {i, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 11 2012, after N. J. A. Sloane *)

The probability that a random pair of elements from the alternating group A_k generates all of A_k is P_k ~ 1-1/k-1/k^2-4/k^3-23/k^4-171/k^5-... = Sum_{n >= 0} a(n)/k^n.
Furthermore, P_k ~ 1 - Sum_{n >= 1} A003319(n)/[k]n, where [k]_n = k(k-1)(k-2)...(k-n+1). Therefore for n >= 2, a(n) = - Sum{i=1..n} A003319(i)*Stirling_2(n-1, i-1). - N. J. A. Sloane.
a(n) ~ -n! / (4 * (log(2))^(n+1)). - Vaclav Kotesovec, Jul 28 2015

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

cp's OEIS Frontend

A113869 Coefficients in asymptotic expansion of probability that a random pair of elements from the alternating group A_k generates all of A_k.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula