|
%I #27 Nov 27 2020 04:07:56
%S 1,-1,-1,-4,-23,-171,-1542,-16241,-194973,-2622610,-39027573,
%T -636225591,-11272598680,-215668335091,-4431191311809,-97316894892644,
%U -2275184746472827,-56421527472282127,-1479397224086870294,-40897073524132164189,-1188896226524012279617
%N Coefficients in asymptotic expansion of probability that a random pair of elements from the alternating group A_k generates all of A_k.
%H Vaclav Kotesovec, <a href="/A113869/b113869.txt">Table of n, a(n) for n = 0..420</a>
%H L. Babai, <a href="http://dx.doi.org/10.1016/0097-3165(89)90068-X">The probability of generating the symmetric group</a>, J. Combin. Theory, A52 (1989), 148-153.
%H J. Bovey and A. Williamson, <a href="https://pdfs.semanticscholar.org/888f/a5eeb2628264ac6822ab3a15bbd2c32d4b1f.pdf">The probability of generating the symmetric group</a>, Bull. London Math. Soc. 10 (1978) 91-96.
%H J. D. Dixon, <a href="http://dx.doi.org/10.1007/BF01110210">The probability of generating the symmetric group</a>, Math. Z. 110 (1969) 199-205.
%H J. D. Dixon, <a href="http://www.combinatorics.org/Volume_12/Abstracts/v12i1r56.html">Asymptotics of Generating the Symmetric and Alternating Groups</a>, Electronic Journal of Combinatorics, vol 11(2), R56.
%H Thibault Godin, <a href="https://arxiv.org/abs/1610.03301">An analogue to Dixon's theorem for automaton groups</a>, arXiv preprint arXiv:1610.03301 [math.GR], 2016.
%H Richard J. Martin, and Michael J. Kearney, <a href="http://dx.doi.org/10.1007/s00493-014-3183-3">Integral representation of certain combinatorial recurrences</a>, Combinatorica: 35:3 (2015), 309-315.
%F 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.
%F 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_.
%F a(n) ~ -n! / (4 * (log(2))^(n+1)). - _Vaclav Kotesovec_, Jul 28 2015
%t 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_ *)
%Y Cf. A003319, A112354, A113871, A114038, A158094, A168246, A306153, A316084.
%K sign,nice
%O 0,4
%A _N. J. A. Sloane_, Jan 26 2006
|
