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Number of degree-n odd permutations of order dividing 4.
+10
6
0, 0, 1, 3, 12, 40, 120, 336, 2128, 13392, 118800, 850960, 6004416, 38408448, 260321152, 1744135680, 17067141120, 167200393216, 1838196972288, 18345298804992, 181218866222080, 1673804042803200, 16992835499329536
FORMULA
E.g.f.: 1/2*exp(x + 1/2*x^2 + 1/4*x^4) - 1/2*exp(x - 1/2*x^2 - 1/4*x^4).
CROSSREFS
Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128, A000704, A061129 - A061132, A048099, A051695, A061133 - A061135, A001465, A061136 - A061140.
Number of cyclic subgroups of prime order in the Alternating Group A_n.
+10
4
0, 0, 1, 7, 31, 121, 526, 2227, 9283, 54931, 694156, 6104011, 76333687, 872550043, 7491293356, 49469173951, 1571562887071, 24729107440927, 584036983443568, 8662243014551731, 87570785839885951, 1147293350653737211, 66175018194591458692, 1378758190497550145383
MATHEMATICA
a[n_] := Sum[If[PrimeQ[p], Sum[If[p > 2 || Mod[k, 2] == 0, n!/(k!*(n - k*p)!*p^k)/(p - 1), 0], {k, 1, n/p}], 0], {p, 2, n}];
PROG
(PARI) a(n)={sum(p=2, n, if(isprime(p), sum(k=1, n\p, if(p>2||k%2==0, n!/(k!*(n-k*p)!*p^k)))/(p-1)))}
Number of degree-n odd permutations of order dividing 6.
+10
3
0, 0, 1, 3, 6, 30, 270, 1386, 6048, 46656, 387180, 2469060, 17204616, 158065128, 1903506696, 18887563800, 163657221120, 2095170230016, 30792968596368, 346564643468976, 3905503235814240, 58609511127871200, 866032039742528736
FORMULA
E.g.f.: exp(x + x^3/3)*sinh(x^2/2 + x^6/6).
Linear recurrence of order 12 whose coefficients are polynomials in n of degree up to 15: see link. - Robert Israel, Jul 13 2018
MAPLE
Egf:= exp(x + x^3/3)*sinh(x^2/2 + x^6/6):
S:= series(Egf, x, 31):
MATHEMATICA
With[{m=30}, CoefficientList[Series[Exp[x + x^3/3]*Sinh[x^2/2 + x^6/6], {x, 0, m}], x]*Range[0, m]!] (* Vincenzo Librandi, Jul 02 2019 *)
PROG
(PARI) my(x='x+O('x^30)); concat([0, 0], Vec(serlaplace( exp(x + x^3/3)*sinh(x^2/2 + x^6/6) ))) \\ G. C. Greubel, Jul 02 2019 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^3/3)*Sinh(x^2/2 + x^6/6) )); [0, 0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Jul 02 2019 (Sage) m = 30; T = taylor(exp(x + x^3/3)*sinh(x^2/2 + x^6/6), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Jul 02 2019
CROSSREFS
Cf. A000085, A001470, A001472, A052501, A053496- A053505, A001189, A001471, A001473, A061121- A061128, A000704, A061129- A061132, A048099, A051695, A061133- A061135, A001465, A061136- A061140.
Weighted sum of all cyclic subgroups of prime order in the Alternating group.
+10
3
0, 0, 3, 18, 90, 390, 2205, 10878, 45318, 256350, 5530305, 55869330, 865551258, 9892489698, 78223384785, 470010394350, 24530527675230, 409760923017198, 10595007772540113, 160826214447439770, 1585844008081570650, 16787211082925012730, 1362379219330719093273
COMMENTS
Sum of p for all p-subgroups in Alt_n.
PROG
(PARI) a(n)={sum(p=2, n, if(isprime(p), sum(k=1, n\p, if(p>2||k%2==0, n!/(k!*(n-k*p)!*p^k)))*p/(p-1)))} \\ Andrew Howroyd, Jul 03 2018
CROSSREFS
Cf. A181951 (number of such subgroups).
Auxiliary sequence for calculation of number of even permutations of degree n and order exactly 2.
+10
2
0, -1, -3, -3, 5, 15, -21, -133, 27, 1215, 935, -12441, -23673, 138047, 469455, -1601265, -9112561, 18108927, 182135007, -161934625, -3804634785, -404007681, 83297957567
REFERENCES
V. Jovovic, Some combinatorial characteristics of symmetric and alternating groups (in Russian), Belgrade, 1980, unpublished.
FORMULA
a(n) = c(n, 2), where c(n, d)=Sum_{k=1..n} (-1)^(k+1)*(n-1)!/(n-k)! *Sum_{l:lcm{k, l}=d} c(n-k, l), c(0, 1)=1.
a(n)=2* A048099(n)- A001189(n)= A048099(n)- A001465(n) a(n)=(-1)^n* A001464(n)-1 a(n)=a(n-1)-(n-1)*(a(n-2)+1) E.g.f.: -e^x+e^(x-(1/2)*x^2) - Matthew J. White (mattjameswhite(AT)hotmail.com), Mar 02 2006 a(n) = Sum((-1)^j*n!/(2^j*j!*(n-2*j)!),j=1..floor(n/2)). - Vladeta Jovovic, Mar 06 2006
Number of degree-n odd permutations of order exactly 4.
+10
1
0, 0, 0, 0, 6, 30, 90, 210, 1680, 12096, 114660, 833580, 5928120, 38112360, 259194936, 1739195640, 17043237120, 167089937280, 1837707369840, 18342985021776, 181206905922720, 1673742164139360, 16992525855006240
FORMULA
E.g.f.: - 1/2*exp(x + 1/2*x^2) + 1/2*exp(x - 1/2*x^2) + 1/2*exp(x + 1/2*x^2 + 1/4*x^4) - 1/2*exp(x - 1/2*x^2 - 1/4*x^4).
CROSSREFS
Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128, A000704, A061129 - A061132, A048099, A051695, A061133 - A061135, A001465, A061136 - A061140.
Number of degree-n odd permutations of order exactly 6.
+10
1
0, 0, 0, 0, 0, 20, 240, 1260, 5600, 45360, 383040, 2451680, 17128320, 157769040, 1902380480, 18882623760, 163633317120, 2095059774080, 30792478993920, 346562329685760, 3905491275514880, 58609449249207360, 866031730098205440
FORMULA
E.g.f.: - 1/2*exp(x + 1/2*x^2) + 1/2*exp(x - 1/2*x^2) + 1/2*exp(x + 1/2*x^2 + 1/3*x^3 + 1/6*x^6) - 1/2*exp(x - 1/2*x^2 + 1/3*x^3 - 1/6*x^6).
CROSSREFS
Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128, A000704, A061129 - A061132, A048099, A051695, A061133 - A061135, A001465, A061136 - A061140.
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