A113869
Coefficients in asymptotic expansion of probability that a random pair of elements from the alternating group A_k generates all of A_k.
Original entry on oeis.org
1, -1, -1, -4, -23, -171, -1542, -16241, -194973, -2622610, -39027573, -636225591, -11272598680, -215668335091, -4431191311809, -97316894892644, -2275184746472827, -56421527472282127, -1479397224086870294, -40897073524132164189, -1188896226524012279617
Offset: 0
- 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.
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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 *)
A168246
Inverse Weigh transform of n!.
Original entry on oeis.org
1, 2, 4, 19, 92, 576, 4156, 34178, 314368, 3199936, 35703996, 433422071, 5687955724, 80256879068, 1211781887796, 19496946568898, 333041104402860, 6019770247224496, 114794574818830716, 2303332661419442569, 48509766592884311132, 1069983257387168051076
Offset: 1
-
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= proc(n) option remember; n! -b(n, n-1) end:
seq(a(n), n=1..30); # Alois P. Heinz, Jun 11 2018
-
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := a[n] = n! - b[n, n - 1];
Array[a, 30] (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz *)
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seq(n)={dirdiv(Vec(log(1+x*Ser(vector(n, n, n!)))), -vector(n, n, (-1)^n/n))} \\ Andrew Howroyd, Jun 22 2018
A305868
Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A001147 (double factorial of odd numbers).
Original entry on oeis.org
1, 2, 12, 87, 816, 9194, 122028, 1859460, 32002076, 613890984, 12989299596, 300556859080, 7550646317520, 204687481289946, 5955892982437120, 185158929516065160, 6125200081143892800, 214837724609502834082, 7963817560398871790604, 311101285877489780292000, 12773912991134665452205048
Offset: 1
1/((1 - x) * (1 - x^2)^2 * (1 - x^3)^12 * (1 - x^4)^87 * (1 - x^5)^816 * ... * (1 - x^n)^a(n) * ...) = 1 + 1*x + 1*3*x^2 + 1*3*5*x^3 + 1*3*5*7*x^4 + ... + A001147(k)*x^k + ...
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nn = 21; f[x_] := Product[1/(1 - x^n)^a[n], {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-k x, 1, {k, 1, nn}]), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
nmax = 20; s = ConstantArray[0, nmax]; Do[s[[j]] = j*(2*j - 1)!! - Sum[s[[d]]*(2*j - 2*d - 1)!!, {d, 1, j - 1}], {j, 1, nmax}]; Table[Sum[MoebiusMu[k/d]*s[[d]], {d, Divisors[k]}]/k, {k, 1, nmax}] (* Vaclav Kotesovec, Aug 09 2019 *)
A305754
Inverse Euler transform of n^n.
Original entry on oeis.org
1, 3, 23, 223, 2800, 42576, 763220, 15734388, 366715248, 9533817400, 273549419552, 8586984241870, 292755986184548, 10772849583399474, 425587711650564816, 17966217346985801150, 807152054953801845760, 38451365602113352159320, 1936082850634342992601636
Offset: 1
(1-x)^(-1) * (1-x^2)^(-3) * (1-x^3)^(-23) * (1-x^4)^(-223) * ... = 1 + x + 4*x^2 + 27*x^3 + 256*x^4 + ... .
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# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> n^n):
seq(a(n), n = 1..19); # Peter Luschny, Nov 21 2022
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n = 20; s = {};
For[i = 1, i <= n, i++, AppendTo[s, i*i^i - Sum[s[[d]]*(i-d)^(i-d), {d, i - 1}]]];
Table[Sum[If[Divisible[i, d], MoebiusMu[i/d], 0]*s[[d]], {d, 1, i}]/i, {i, n}] (* Jean-François Alcover, May 10 2019 *)
A305786
Inverse Euler transform of (-1)^n * n!.
Original entry on oeis.org
-1, 2, -4, 17, -92, 576, -4156, 34159, -314368, 3199936, -35703996, 433421495, -5687955724, 80256879068, -1211781887796, 19496946534720, -333041104402860, 6019770247224496, -114794574818830716, 2303332661416242633, -48509766592884311132, 1069983257387168051076
Offset: 1
(1-x) * (1-x^2)^(-2) * (1-x^3)^4 * (1-x^4)^(-17) * ... = 1 - x + 2*x^2 - 6*x^3 + 24*x^4 - ... .
A380498
Inverse Euler transform of primorial numbers.
Original entry on oeis.org
2, 3, 20, 150, 1860, 24950, 444060, 8583780, 202071920, 5992771854, 186947632200, 7001535703840, 288868991951760, 12455290280427090, 587972068547997856, 31327583556941402160, 1856116108295418943020, 113366872636395265380920, 7619343577986975410930880, 541957669076266398658079700
Offset: 1
- Eric Weisstein's World of Mathematics, Primorial.
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p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
ietr:= proc(p) uses numtheory; (c-> proc(n) option remember;
`if`(n=0, 1, add(mobius(n/d)*c(d), d=divisors(n))/n) end)(
proc(n) option remember; n*p(n)-add(thisproc(j)*p(n-j), j=1..n-1) end)
end:
a:= ietr(p):
seq(a(n), n=1..20); # Alois P. Heinz, Jan 25 2025
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primorial[n_] := Product[Prime[j], {j, 1, n}]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i] + j - 1, j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := primorial[n] - b[n, n - 1]; a /@ Range[20]
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