A246945 -id:A246945 - cp's OEIS frontend

A003483 Number of square permutations of n elements.

1, 1, 1, 3, 12, 60, 270, 1890, 14280, 128520, 1096200, 12058200, 139043520, 1807565760, 22642139520, 339632092800, 5237183952000, 89032127184000, 1475427973219200, 28033131491164800, 543494606861606400, 11413386744093734400, 235075995738558374400, 5406747901986842611200, 126214560713084056012800

Offset: 0

N. J. A. Sloane

Number of permutations p in S_n such that there exists q in S_n with q^2=p.

"A permutation P has a square root if and only if the numbers of cycles of P that have each even length are even numbers." [Theorem 4.8.1. on p.147 from the Wilf reference]. - Joerg Arndt, Sep 08 2014

			a(3) = 3: permutations with square roots are identity and two 3-cycles.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.11.
H. S. Wilf, Generatingfunctionology, 3rd ed., A K Peters Ltd., Wellesley, MA, 2006, p. 157.

Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 101 terms from N. J. A. Sloane)
Edward A. Bender, Asymptotic methods in enumeration, SIAM Review 16 (1974), no. 4, p. 509.
Joseph Blum, Letters to N. J. A. Sloane, 1974
J. Blum, Enumeration of the square permutations in S_n, J. Combin. Theory, A 17 (1974), 156-161.
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 36.
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
P. Flajolet et al., A hybrid of Darboux's method and singularity analysis in combinatorial asymptotics, arXiv:math.CO/0606370, p. 18, Proposition 2.
Yuewen Luo, Counting Permutations in S_{2n} and S_{2n+1}, arXiv:2407.07366 [math.CO], 2024.
M. R. Pournaki, On the number of even permutations with roots, The Australasian Journal of Combinatorics, Volume 45, 2009, pp. 37-42.
N. Pouyanne, On the number of permutations admitting an m-th root, Electron. J. Combin., 9 (2002), #R3.
Bob Smith and N. J. A. Sloane, Correspondence, 1979
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 148, Eq. 4.8.1.

Cf. A103619 (cube root), A103620 (fourth root), A215716 (fifth root), A215717 (sixth root), A215718 (seventh root).
Column k=2 of A247005.
Cf. A246945, A247621.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(i, 2)=0 and irem(j, 2)=1, 0, (i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 08 2014

max = 20; f[x_] := Sqrt[(1 + x)/(1 - x)]*  Product[ Cosh[x^(2*k)/(2*k)], {k, 1, max}]; se = Series[ f[x], {x, 0, max}]; CoefficientList[ se, x]*Range[0, max]! (* Jean-François Alcover, Oct 05 2011, after g.f. *)
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[Mod[i, 2] == 0 && Mod[j, 2] == 1, 0, (i-1)!^j* multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 23 2015, after Alois P. Heinz *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Sqrt[ (1 + x) / (1 - x)] Product[ Cosh[ x^k / k], {k, 2, n, 2}], {x, 0, n}]]; (* Michael Somos, Jul 11 2018 *)

N=66; x='x+O('x^66);
Vec(serlaplace( sqrt((1+x)/(1-x))*prod(k=1,N, cosh(x^(2*k)/(2*k)))))
\\ Joerg Arndt, Sep 08 2014

E.g.f.: sqrt((1 + x)/(1 - x)) * Product_{k>=1} cosh( x^(2*k)/(2*k) ). [Blum, corrected].
a(2*n+1) = (2*n + 1)*a(2*n).
Asymptotics: a(n) ~ n! * sqrt(2/(n*Pi)) * e^G, where e^G = Product_{k>=1} cosh(1/(2k)) ~ 1.22177951519253683396485298445636121278881... (see A246945). - corrected by Vaclav Kotesovec, Sep 13 2014
G = Sum_{j>=1} (-1)^(j + 1) * Zeta(2*j)^2 * (1 - 1/2^(2*j)) / (j * Pi^(2*j)). - Vaclav Kotesovec, Sep 20 2014

More terms from Vladeta Jovovic, Mar 28 2001
Additional comments from Michael Somos, Jun 27 2002
Minor edits by Vaclav Kotesovec, Sep 16 2014 and Sep 21 2014

A249673 Decimal expansion of Product_{n>=1} cosh(1/n).

Original entry on oeis.org

2, 1, 1, 6, 4, 6, 5, 5, 3, 6, 5, 0, 5, 4, 8, 4, 7, 7, 5, 8, 7, 8, 5, 7, 2, 2, 2, 7, 0, 2, 5, 8, 3, 1, 9, 8, 8, 1, 4, 8, 0, 8, 9, 3, 9, 2, 8, 0, 9, 0, 8, 2, 5, 6, 8, 2, 8, 1, 3, 4, 8, 0, 7, 8, 6, 9, 4, 2, 3, 8, 3, 0, 7, 2, 8, 9, 0, 1, 1, 7, 2, 9, 9, 6, 1, 9, 3, 4, 6, 5, 9, 2, 4, 3, 1, 0, 8, 8, 9, 4, 2, 8, 6, 3, 7

Offset: 1

Vaclav Kotesovec, Nov 03 2014

			2.116465536505484775878572227025831988148089392809082568281348...

Cf. A118817, A230821, A051762, A218504, A246945, A270614.

Cf. A027641, A027642.

evalf(exp(sum(log(cosh(1/n)), n=1..infinity)), 100)

default(realprecision,120); exp(sumpos(k=1, log(cosh(1/k))))

From Amiram Eldar, Jul 30 2023: (Start)
Equals exp(Sum_{k>=1} 2^(2*k-1)*(2^(2*k)-1)*B(2*k)*zeta(2*k)/(k*(2*k)!)), where B(k) is the k-th Bernoulli number.
Equals exp(Sum_{k>=1} (-1)^(k+1)*(2^(2*k)-1)*zeta(2*k)^2/(k*Pi^(2*k))). (End)

A118817 Decimal expansion of Product_{n >= 1} cos(1/n).

Original entry on oeis.org

3, 8, 8, 5, 3, 6, 1, 5, 3, 3, 3, 5, 1, 7, 5, 8, 5, 9, 1, 8, 4, 3, 2, 9, 5, 7, 5, 6, 8, 7, 0, 3, 5, 9, 0, 5, 0, 1, 3, 9, 0, 0, 5, 2, 8, 5, 9, 7, 5, 1, 7, 9, 2, 1, 9, 1, 3, 1, 8, 4, 6, 1, 1, 9, 9, 8, 7, 9, 8, 7, 4, 9, 4, 3, 4, 6, 3, 3, 9, 3, 2, 7, 6, 8, 3, 8, 8, 4, 3, 1, 9, 7, 8, 1, 3, 8, 3, 4, 0, 8, 2, 2, 4, 1, 3

Offset: 0

Fredrik Johansson, May 23 2006

			0.38853615333517585918432957568703590501390...

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A046990, A046991.
Cf. A051762, A085365, A246945, A230821, A249673.
Cf. A027641, A027642.

nn:= 120:
p:= product(cos(1/n), n=1..infinity):
f:= evalf(p, nn+10):
s:= convert(f, string):
seq(parse(s[n+1]), n=1..nn);  # Alois P. Heinz, Nov 04 2013

S = Series[Log[Cos[x]], {x, 0, 400}]; N[Exp[N[Sum[SeriesCoefficient[S, 2k] Zeta[2k], {k, 1, 200}], 70]], 50]
Block[{$MaxExtraPrecision = 1000}, Do[Print[N[1/Exp[Sum[(2^(2*n) - 1)*Zeta[2*n]^2/(n*Pi^(2*n)), {n, 1, m}]], 110]], {m, 250, 300}]] (* Vaclav Kotesovec, Sep 20 2014 *)

exp(-sumpos(n=1,-log(cos(1/n)))) \\ warning: requires 2.6.2 or greater; Charles R Greathouse IV, Nov 04 2013

T(n)=((-4)^n-(-16)^n)*bernfrac(2*n)/2/n/(2*n)!
lm=lambertw(2*log(Pi/2)*10^default(realprecision))/2/log(Pi/2); exp(-sum(n=1,lm,T(n)*zeta(2*n))) \\ Charles R Greathouse IV, Nov 06 2013

Equals exp(Sum_{n>=1} -c(n)*zeta(2*n)), where c(n) = A046990(n)/A046991(n).
Equals exp(-Sum_{n>=1} (2^(2*n)-1) * Zeta(2*n)^2 / (n*Pi^(2*n)) ). - Vaclav Kotesovec, Sep 20 2014
Equals exp(Sum_{k>=1} (-1)^k*2^(2*k-1)*(2^(2*k)-1)*B(2*k)*zeta(2*k)/(k*(2*k)!)), where B(k) is the k-th Bernoulli number. - Amiram Eldar, Jul 30 2023

Corrected offset and extended by Robert G. Wilson v, Nov 03 2013

A270614 Decimal expansion of Product_{k>=1} ((1 + sinh(1/k)) / exp(1/k)).

Original entry on oeis.org

6, 2, 5, 6, 3, 5, 8, 0, 1, 9, 7, 7, 9, 4, 9, 8, 4, 4, 5, 9, 7, 4, 8, 4, 8, 1, 6, 1, 0, 3, 1, 0, 4, 4, 2, 9, 9, 0, 5, 2, 0, 6, 3, 5, 5, 1, 1, 2, 7, 1, 3, 1, 9, 6, 8, 9, 6, 0, 3, 4, 9, 8, 3, 2, 5, 9, 8, 5, 8, 6, 7, 6, 4, 3, 6, 4, 4, 1, 0, 3, 5, 4, 3, 6, 7, 5, 6, 0, 7, 1, 2, 4, 5, 6, 2, 7, 5, 6, 4, 3, 1, 3, 1, 8, 1, 8

Offset: 0

Vaclav Kotesovec, Mar 20 2016

			0.625635801977949844597484816103104429905206355112713196896034983259858...

Cf. A130263, A246945, A249673, A270597.

evalf(Product((1+sinh(1/k))/exp(1/k), k=1..infinity), 120);

default(realprecision, 120); exp(sumpos(k=1, log((1+sinh(1/k))/exp(1/k))))

Equals limit n->infinity A130263(n)/n!.

A246948 Decimal expansion of the coefficient c appearing in the asymptotic expression of the probability that a random n-permutation is a cube as c/n^3.

Original entry on oeis.org

1, 0, 7, 2, 9, 9, 7, 9, 4, 4, 3, 8, 9, 5, 2, 7, 0, 1, 7, 7, 3, 7, 9, 7, 1, 3, 9, 4, 9, 5, 4, 4, 6, 5, 5, 5, 5, 6, 8, 1, 0, 1, 4, 1, 8, 8, 3, 1, 4, 3, 7, 4, 0, 4, 5, 6, 6, 7, 8, 5, 3, 5, 2, 4, 3, 1, 9, 7, 8, 4, 5, 0, 2, 2, 4, 8, 6, 2, 7, 2, 8, 1, 1, 9, 6, 2, 1, 9, 2, 6, 4, 1, 1, 9, 5, 7, 0, 3, 4, 1, 5, 5, 9, 2, 7

Offset: 1

Jean-François Alcover, Sep 08 2014

			1.072997944389527017737971394954465555681...

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 36.

Cf. A246945.

evalf(3^(5/6) * GAMMA(1/3) / (2*Pi) * Product(1/3*(exp(1/(3*k)) + 2*exp(-1/(6*k)) * cos(sqrt(3)/(6*k))), k=1..infinity),100) # Vaclav Kotesovec, Sep 17 2014

digits = 40; m0 = 1000; dm = 1000; psi[x_] := 1/3*(E^x + 2*E^(-x/2)*Cos[Sqrt[3]*(x/2)]); tail[m_] := (-98761420800*PolyGamma[2, m] - 4572288*PolyGamma[5, m] - 53*PolyGamma[8, m])/31998700339200; Clear[f]; f[m_] := f[m] = Sum[Log[psi[1/(3*k)]], {k, 1, m - 1}] + tail[m] // N[#, digits + 10] &; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits + 5] != RealDigits[f[m - dm], 10, digits + 5], Print["f(", m, ") = ", f[m]]; m = m + dm]; c = 3^(5/6)*Gamma[1/3]/(2*Pi)*E^f[m]; RealDigits[c, 10, 40] // First

default(realprecision,150); 3^(5/6) * gamma(1/3) / (2*Pi) * exp(sumpos(k=1,log(1/3*(exp(1/(3*k)) + 2*exp(-1/(6*k)) * cos(sqrt(3)/(6*k)))))) \\ Vaclav Kotesovec, Sep 21 2014

c = 3^(5/6)*Gamma(1/3)/(2*Pi)*prod_{k>=1} psi(1/(3k)), where psi(x) = 1/3*(e^x + 2*e^(-x/2)*cos(sqrt(3)*(x/2))).

More terms from Vaclav Kotesovec, Sep 17 2014

A349645 Triangular array read by rows: T(n,k) is the number of square n-permutations possessing exactly k cycles; n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 11, 0, 1, 0, 24, 0, 35, 0, 1, 0, 0, 184, 0, 85, 0, 1, 0, 720, 0, 994, 0, 175, 0, 1, 0, 0, 9708, 0, 4249, 0, 322, 0, 1, 0, 40320, 0, 72764, 0, 14889, 0, 546, 0, 1, 0, 0, 648576, 0, 402380, 0, 44373, 0, 870, 0, 1

Offset: 0

Steven Finch, Nov 23 2021

A permutation p in S_n is a square if there exists q in S_n with q^2=p.

For such a p, the number of cycles of any even length in its disjoint cycle decomposition must be even.

			The three square 3-permutations are (1, 2, 3) with three cycles (fixed points) and (3, 1, 2) & (2, 3, 1), each with one cycle.
Among the twelve square 4-permutations are {1, 4, 2, 3} & {1, 3, 4, 2} and {3, 4, 1, 2} & {4, 3, 2, 1}, all with two cycles but differing types.
Triangle begins:
[0]   1;
[1]   0,   1;
[2]   0,   0,    1;
[3]   0,   2,    0,   1;
[4]   0,   0,   11,   0,    1;
[5]   0,  24,    0,  35,    0,   1;
[6]   0,   0,  184,   0,   85,   0,   1;
[7]   0, 720,    0, 994,    0, 175,   0,   1;
[8]   0,   0, 9708,   0, 4249,   0, 322,   0,   1;
...

Alois P. Heinz, Rows n = 0..200, flattened
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.

Columns k=0-1 give: A000007, A005359(n-1).
Row sums give A003483.
T(n+2,n) gives A000914.
Cf. A214851, A246945.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(i, 2)=0 and irem(j, 2)=1, 0, (i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1))*x^j, j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Nov 23 2021

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0,
     Sum[If[Mod[i, 2] == 0 && Mod[j, 2] == 1, 0, (i-1)!^j*multinomial[n,
     Join[{n-i*j}, Table[i, {j}]]]/j!*b[n-i*j, i-1]]*x^j, {j, 0, n/i}]]]];
T[n_] := With[{p = b[n, n]}, Table[Coefficient[p, x, i], {i, 0, n}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

cp's OEIS Frontend

A003483 Number of square permutations of n elements.

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A249673 Decimal expansion of Product_{n>=1} cosh(1/n).

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A118817 Decimal expansion of Product_{n >= 1} cos(1/n).

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A270614 Decimal expansion of Product_{k>=1} ((1 + sinh(1/k)) / exp(1/k)).

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A246948 Decimal expansion of the coefficient c appearing in the asymptotic expression of the probability that a random n-permutation is a cube as c/n^3.

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A349645 Triangular array read by rows: T(n,k) is the number of square n-permutations possessing exactly k cycles; n >= 0, 0 <= k <= n.

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