A113871 -id:A113871 - 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

A114038 Analog of A113869 for three generators.

Original entry on oeis.org

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

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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

A316862 Expansion of 1/(Sum_{k>=0} (k!)^3 x^k).

Original entry on oeis.org

1, -1, -7, -201, -13351, -1697705, -369575303, -127249900617, -65286578868455, -47651775381867241, -47688241963081263175, -63505249400026210723209, -109775495351620406817045415, -241236985075124408660287423529, -662075390371447206867029299628807

Offset: 0

Seiichi Manyama, Jul 15 2018

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Seiichi Manyama, Table of n, a(n) for n = 0..181

1/(Sum_{k>=0} (k!)^b x^k): A167894 (b=1), A113871 (b=2), this sequence (b=3).

Cf. A000442.

a[n_] := -Sum[(k!)^3*a[n - k], {k, n}]; a[0] = 1; Array[a, 15, 0] (* Robert G. Wilson v, Jul 15 2018 *)
nmax = 20; CoefficientList[Series[1/Sum[k!^3 * x^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 08 2020 *)

a(0) = 1, a(n) = -Sum_{k=1..n} (k!)^3 * a(n-k).

a(n) ~ -(n!)^3 * (1 - 2/n^3 - 13/n^6 - 39/n^7 - 78/n^8 - 518/n^9 - 3687/n^10 - ...). - Vaclav Kotesovec, Dec 08 2020

A306629 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/(Sum_{j>=0} (j!)^k * x^j).

Original entry on oeis.org

1, 1, -1, 1, -1, 0, 1, -1, -1, 0, 1, -1, -3, -3, 0, 1, -1, -7, -29, -13, 0, 1, -1, -15, -201, -499, -71, 0, 1, -1, -31, -1265, -13351, -13101, -461, 0, 1, -1, -63, -7713, -328975, -1697705, -486131, -3447, 0, 1, -1, -127, -46529, -7946143, -206659569, -369575303, -24266797, -29093, 0

Offset: 0

Seiichi Manyama, Mar 02 2019

			Square array begins:
    1,    1,       1,          1,             1, ...
   -1,   -1,      -1,         -1,            -1, ...
    0,   -1,      -3,         -7,           -15, ...
    0,   -3,     -29,       -201,         -1265, ...
    0,  -13,    -499,     -13351,       -328975, ...
    0,  -71,  -13101,   -1697705,    -206659569, ...
    0, -461, -486131, -369575303, -268312660751, ...

Seiichi Manyama, Antidiagonals n = 0..59, flattened

Columns 1-3 give A167894, A113871, A316862.
Rows 0-2 give A000012, (-1)*A000012, (-1)*A000225.
Main diagonal gives A306630.

A(0,k) = 1 and A(n,k) = -Sum_{j=1..n} (j!)^k * A(n-j,k) for n > 0.

A114039 Expansion of g.f. 1/Sum_{k>=0} k!(k!+1)x^k/2.

Original entry on oeis.org

1, -1, -2, -16, -257, -6613, -243980, -12155224, -786429623, -64166829085, -6449457120962, -783277824629140, -113105715390915929, -19155829563735891169, -3760880464339217383124, -847344181950467763488740, -217159542960607254804746279, -62816327344923649504400271697

Offset: 0

N. J. A. Sloane, Feb 01 2006

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Cf. A113871.

nmax=17; CoefficientList[Series[1/Sum[k!*(k!+1)*x^k/2,{k,0,nmax}],{x,0,nmax}],x] (* Stefano Spezia, Oct 09 2023 *)

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.

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A114038 Analog of A113869 for three generators.

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A316862 Expansion of 1/(Sum_{k>=0} (k!)^3 x^k).

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A306629 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/(Sum_{j>=0} (j!)^k * x^j).

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A114039 Expansion of g.f. 1/Sum_{k>=0} k!*(k!+1)*x^k/2.

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A114039 Expansion of g.f. 1/Sum_{k>=0} k!(k!+1)x^k/2.