A316084 -id:A316084 - 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

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

A363255 Product_{n>=1} 1 / (1 - a(n)x^n) = 1 + 1!!x + 3!!x^2 + 5!!x^3 + 7!!*x^4 + ...

Original entry on oeis.org

1, 2, 12, 86, 816, 9126, 122028, 1855802, 32001504, 613558458, 12989299596, 300515004558, 7550646317520, 204680035934550, 5955892801274796, 185157207502788074, 6125200081143892800, 214837212308039658666, 7963817560398871790604, 311101097650387613661510

Offset: 1

Ilya Gutkovskiy, May 23 2023

nonn

Cf. A001147, A305868, A305870, A316084, A363254.

A[m_, n_] := A[m, n] = Which[m == 1, (2 n - 1)!!, m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m - 1, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 20]

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

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A363255 Product_{n>=1} 1 / (1 - a(n)x^n) = 1 + 1!!x + 3!!x^2 + 5!!x^3 + 7!!*x^4 + ...

Views

Author

Keywords

Crossrefs

Programs

Mathematica

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

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A363255 Product_{n>=1} 1 / (1 - a(n)*x^n) = 1 + 1!!*x + 3!!*x^2 + 5!!*x^3 + 7!!*x^4 + ...

Views

Author

Keywords

Crossrefs

Programs

Mathematica

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

A363255 Product_{n>=1} 1 / (1 - a(n)x^n) = 1 + 1!!x + 3!!x^2 + 5!!x^3 + 7!!*x^4 + ...