A057863 -id:A057863 - cp's OEIS frontend

A279886 a(n) = A057863(n+1) * Sum_{k=0..n}(k! / (2*k+1)!!).

1, 4, 66, 7200, 6917400, 72442188000, 9822893856930000, 19942863749656848000000, 687732249521129504041350000000, 450437284567157389148103391935000000000, 6194243041031315772374678081343893262937500000000

Offset: 0

Daniel Suteu, Dec 21 2016

Daniel Suteu, Table of n, a(n) for n = 0..29

Cf. A057863.

Table[Product[(2 k - 1)!!, {k, n + 1}] Sum[(j!/(2 j + 1)!!), {j, 0, n}], {n, 0, 10}] (* Michael De Vlieger, Dec 22 2016 *)

a(n) = prod(k=0,n,prod(i=0, k, 2*i+1))*sum(k=0,n,k!/prod(i=0,k,2*i+1));

a(n) ~ Pi/2 * A057863(n+1).

a(0) = 1, a(n) = a(n-1) * (2*n+1)!! + n! * A057863(n).

A005118 Number of simple allowable sequences on 1..n containing the permutation 12...n.

Original entry on oeis.org

1, 1, 1, 2, 16, 768, 292864, 1100742656, 48608795688960, 29258366996258488320, 273035280663535522487992320, 44261486084874072183645699204710400, 138018895500079485095943559213817088756940800

Offset: 0

N. J. A. Sloane

For n >= 2 by the hook length formula a(n) is also the number of Young tableaux of size 1+2+...+(n-1) = n*(n-1)/2 that correspond to the partition (1,2,...n-1), i.e., triangular Young tableaux. For example, for n=5 the shape of the tableau is xxxx / xxx / xx / x. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 04 2001
Also, a(n) is the degree of the symplectic Grassmannian, the projective variety of all maximal isotropic subspaces in a complex vector space of dimension 2n-2 with a symplectic form. See Hiller's paper. - Burt Totaro (b.totaro(AT)dpmms.cam.ac.uk), Oct 29 2002
Also, for n >= 2, a(n) is the number of maximal chains in the poset of Dyck paths ordered by inclusion. - Jennifer Woodcock (Jennifer.Woodcock(AT)ugdsb.on.ca), May 21 2008
a(n) is the number of minimal decompositions of the "flip" permutation n(n-1)..21 in terms of the n-1 standard Coxeter generators (i i+1) ("reduced decompositions", cf. Stanley). As such, it is also the number of positive n-strand braid words representing the Garside braid Delta(n) (the half-turn) (cf. Epstein's book, lemma 9.1.14). - Maxime Bourrigan, Apr 04 2011
For n >= 1, the normalized volume of the subpolytope of the Birkhoff polytope obtained by taking the convex hull of all (2n)x(2n) permutation matrices corresponding to alternating permutations that also avoid the pattern 123. - Robert Davis, Dec 04 2016

D. B. A. Epstein with J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson and W. P. Thurston, Word Processing in Groups, Jones and Bartlett Publishers, Boston, MA, 1992. xii+330 pp.
J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 102.
G. Kreweras, Sur un problème de scrutin à plus de deux candidats, Publications de l'Institut de Statistique de l'Université de Paris, 26 (1981), 69-87.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..40
Omer Angel, Alexander E. Holroyd, Dan Romik, and Balint Virag, Random Sorting Networks, arXiv preprint arXiv:0609538 [math.PR], 2006.
Joerg Arndt, The a(4)=16 Young tableaux of shape [3, 2, 1].
Sara C. Billey and Peter R. W. McNamara, The contributions of Stanley to the fabric of symmetric and quasisymmetric functions, arXiv preprint, 2015.
Tobias Boege, Alessio D'Alì, Thomas Kahle, Bernd Sturmfels, The Geometry of Gaussoids, arXiv:1710.07175 [math.CO], 2017.
R. Davis and B. Sagan, Pattern-Avoiding Polytopes, 2016
FindStat - Combinatorial Statistic Finder, The number of ways to write a permutation as a minimal length product of simple transpositions
M. J. Hay, J. Schiff, and N. J. Fisch, Maximal energy extraction under discrete diffusive exchange, arXiv preprint arXiv:1508.03499 [physics.plasm-ph], 2015.
H. Hiller, Combinatorics and intersection of Schubert varieties, Comment. Math. Helv. 57 (1982), 41-59.
D. Kim, Finding k Shortest Paths in Cayley Graphs of Finite Groups, Graphs and Combinatorics 40, 120 (2024). See Formula at p. 13.
G. Kreweras, Sur un problème de scrutin à plus de deux candidats, Publications de l'Institut de Statistique de l'Université de Paris, 26 (1981), 69-87. [Annotated scanned copy]
Joshua Maglione and Christopher Voll, Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration, arXiv:2410.08075 [math.CO], 2024. See pp. 34, 39.
R. P. Stanley, A combinatorial miscellany
R. P. Stanley, Ordering events in Minkowski space, arXiv:math/0501256 [math.CO], 2005.
R. P. Stanley, On the number of reduced decompositions of elements of Coxeter groups, European J. Combin., 5 (1984), 359-372.

Cf. A003121, A018241, A057863, A246865, A289778.

A005118 := proc(n) local i; binomial(n,2)!/product( (2*i+1)^(n-i-1), i=0..n-2 ); end;

Table[Binomial[n, 2]!/Product[(2*i + 1)^(n - i - 1), {i, 0, n - 2}], {n, 0, 10}] (* T. D. Noe, May 29 2012 *)

a(n) = C(n, 2)!/(1^{n-1} * 3^{n-2} *...* (2n-3)^1 ).
a(n) = (n*(n-1)/2)!/A057863(n-1) (n>=1). - Emeric Deutsch, May 21 2004
a(n) = A153452(A002110(n-1)). - Naohiro Nomoto, Jan 01 2009
From Alois P. Heinz, Nov 18 2012: (Start)
a(n+1) = A219272(A000217(n),n) = A219274(A000217(n),n) = A219311(A000217(n),n).
a(n) = A193536(n,A000217(n-1)) = A193629(n,A000217(n-1)). (End)
a(n) ~ sqrt(Pi) * n^(n^2/2-n/2+23/24) * exp(n^2/4-n/2+7/24) / (A^(1/2) * 2^(n^2-n/2-7/24)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 13 2014

Citation corrected by Matthew J. Samuel, Feb 01 2011

A098694 Double-superfactorials: a(n) = Product_{k=1..n} (2k)!.

Original entry on oeis.org

1, 2, 48, 34560, 1393459200, 5056584744960000, 2422112183371431936000000, 211155601241022491077587763200000000, 4417964278440225627098723475313498521600000000000

Offset: 0

Ralf Stephan, Sep 22 2004

nonn

Hankel transform of double factorial numbers A001147. - Paul Barry, Jan 28 2008

Hankel transform of A112934(n+1). - Paul Barry, Dec 04 2009

Seiichi Manyama, Table of n, a(n) for n = 0..28
C. Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.

Cf. A000178, A010050, A074962, A168467, A268504, A268505, A268506, A271946, A271947.

[&*[ Factorial(2*k): k in [0..n] ]: n in [0..10]]; // Vincenzo Librandi, Dec 11 2016

Table[Product[(2k)!,{k,1,n}],{n,0,10}] (* Vaclav Kotesovec, Nov 13 2014 *)

a(n) = prod(k=1, n, (2*k)!); \\ Michel Marcus, Dec 11 2016

from math import prod
def A098694(n): return prod(((k+1)*((k<<1)+1)<<1)**(n-k) for k in range(1,n+1))<Chai Wah Wu, Nov 26 2023

a(n) = Product_{k=0..n} (2*(k+1)*(2*k+1))^(n-k). - Paul Barry, Jan 28 2008
a(n) = A000178(n)*A057863(n)*A006125(n+1) = A121835(n)*A006125(n+1). - Paul Barry, Jan 28 2008
G.f.: G(0)/(2*x)-1/x, where G(k)= 1  + 1/(1 - 1/(1 + 1/(2*k+2)!/x/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
a(n) ~ 2^(n^2+2*n+17/24) * n^(n^2+3*n/2+11/24) * Pi^((n+1)/2) / (A^(1/2) * exp(3*n^2/2+3*n/2-1/24)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 13 2014
a(n) = A^(3/2)*2^(n^2+n-1/24)*Pi^(-n/2-1/4)*G(n+3/2)*G(n+2)/exp(1/8), where G(n) is the Barnes G-function and A is the Glaisher-Kinkelin constant. - Ilya Gutkovskiy, Dec 11 2016
a(n) = A000178(2*n + 1) / A168467(n). - Vaclav Kotesovec, Oct 28 2017
For n > 0, a(n) = 2^((n+1)/2) * n * sqrt(BarnesG(2*n)*Gamma(n)) * Gamma(2*n). - Vaclav Kotesovec, Nov 27 2024

A092042 Decimal expansion of e^(1/4).

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Mar 27 2004

e^(1/4) is also the integral from 0 to infinity of e^(-x) * I_0(sqrt(x)), where I_0(z) is a modified Bessel function. - Jean-François Alcover, Mar 10 2011

e^(1/4) maximizes the value of x^(c/(x^4)) for any real positive constant c, and minimizes for it for a negative constant, on the range x > 0. - A.H.M. Smeets, Aug 16 2018

			1.28402541668774148407342056806243645833....

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
D. M. Bătinetu-Giurgiu, Problem 4133, Crux Mathematicorum, Vol. 42, No. 4 (2016), p. 174; Solution to Problem 4133, ibid., Vol. 43, No. 4 (2017), pp. 167-169.
Index entries for transcendental numbers

Cf. A001113, A019774, A057863, A078688, A092426.

evalf(exp(1/4)); # Muniru A Asiru, Aug 16 2018

RealDigits[(E)^(1/4), 10, 100][[1]] (* Vincenzo Librandi, Mar 01 2013 *)

PARI
```
exp(1/4) \\ Michel Marcus, Jan 19 2017
```

e^(1/4) = 1/2*( 1 +(5 +(9 +(13 +...)/12)/8)/4 ) = 1 +(1 +(1 +(1 +...)/12)/8)/4. - Rok Cestnik, Jan 19 2017
Equals lim_{n->oo} ((2*n-1)!!)^(1/(2*n))/A057863(n)^(1/n^2) (Bătinetu-Giurgiu, 2016). - Amiram Eldar, Apr 10 2022
Equals (Integral_{x=1..oo} 1/(x*log(x)^log(log(x))) dx)/sqrt(Pi). - Kritsada Moomuang, Jun 03 2025

A086205 Determinant of n X n matrix M_(i,j) = binomial(i^2, j).

Original entry on oeis.org

1, 1, 6, 360, 302400, 4572288000, 1520925880320000, 13153940405570764800000, 3412910854477066178396160000000, 30107378079113824305786648526848000000000

Offset: 0

Benoit Cloitre, Aug 27 2003

Closed form can be deduced from the conjecture that the count of triangular semi-standard Young tableaux with shape (n,...,1) and max part n equals 2^((n-1)n/2); see Mathematica line. - Wouter Meeussen, Nov 26 2017

Cf. A057863, A266091.

Table[Product[(2*k-1)!/(k-1)!,{k,1,n}],{n,0,10}] (* Vaclav Kotesovec, Jul 11 2015 *)
Table[Product[ -i+j+n, {i,n}, {j, 1-i+n}], {n,0,10}];
Round[Table[Sqrt[Glaisher]/(2^(1/24 - n^2)* E^(1/24 + Derivative[1, 0][Zeta][-1, 1/2 + n])* Pi^(1/4 + n/2)*Gamma[1/2 + n]^(-(1/2) - n)), {n, 16}]] (* see comments *) (* Wouter Meeussen, Nov 26 2017 *)
Table[Glaisher^(3/2) * 2^(n^2 - 1/24) * BarnesG[n + 3/2] / (E^(1/8) * Pi^(n/2 + 1/4)), {n, 0, 12}] (* Vaclav Kotesovec, Mar 24 2019 *)

PARI
```
a(n)=prod(k=1,n,(2*k-1)!/(k-1)!)
```

a(n)=matdet(matrix(n,n,i,j,binomial(i^2,j)))

a(n) = Product_{k = 1..n} (2*k-1)!/(k-1)!.
a(n) ~ A^(1/2) * 2^(n^2 + n/2 + 5/24) * n^(n^2/2 + n/2 + 1/24) / exp(3*n^2/4 + n/2 + 1/24), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 11 2015
a(n) = s_lambda(1,2,3,...,n) where s is the Schur polynomial in n variables and lambda is the partition (n,n-1,n-2,...,1). - Leonid Bedratyuk, Feb 06 2022
From Peter Bala, Feb 19 2023: (Start)
a(n) = Product_{k = 0..n} (n + k)!/(2*k)!. Cf. A266091.
a(n) = 2^(n*(n-1)/2) * A057863(n).
a(n+1)*a(n-1) = (4*n + 2)*a(n)^2.
a(n)^3*a(n+2) + a(n-1)*a(n+1)^3 = (8*n + 8)*(a(n)*a(n + 1))^2 for n >= 1.
Conjecture: a(n) = the determinant of the n X n matrix ( binomial(i^2 + z, j) ) for arbitrary complex z. (End)

A263416 a(n) = Product_{k=0..n} (3*k+1)^(n-k).

Original entry on oeis.org

1, 1, 4, 112, 31360, 114150400, 6648119296000, 7356542888181760000, 179090034163620983603200000, 108995627512253039588776345600000000, 1857397104331364341705287836001894400000000000, 981210407605679794692064339146706741991833600000000000000

Offset: 0

Vaclav Kotesovec, Oct 17 2015

G. C. Greubel, Table of n, a(n) for n = 0..50
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant
Eric Weisstein's World of Mathematics, Polygamma Function
Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function.

Cf. A057863, A074962, A252798, A261024, A263405, A263414, A263417, A369468.

A263416:=n->mul((3*k+1)^(n-k), k=0..n): seq(A263416(n), n=0..11); # Wesley Ivan Hurt, Nov 12 2015

Table[Product[(3*k+1)^(n-k),{k,0,n}],{n,0,12}] (* or *)
Table[1/FullSimplify[(Gamma[1/3]^((v-1)/3) / 3^((v-1)/18)) * Exp[Integrate[(E^((3-v)*x) - E^(2*x))/(x*(E^(3*x)-1)^2) + (v-1) * (1/(3*x*(E^(3*x)-1)) + 1/(6*x*E^(3*x)) - (v+1)/(18*x*E^x)), {x, 0, Infinity}]]], {v, 1, 34, 3}]
Round@Table[3^(n(n+1)/2) BarnesG[n+4/3]/(BarnesG[1/3] Gamma[1/3]^(n+1)), {n, 0, 12}] (* Vladimir Reshetnikov, Nov 11 2015 *)

a(n) = prod(k=0, n, (3*k+1)^(n-k)); \\ Michel Marcus, Nov 12 2015

a(n) ~ A^(1/3) * 2^(n/2 + 1/6) * 3^(n^2/2 + n/2 - 1/72) * n^(n^2/2 + n/3 - 1/36) * Pi^(n/2 + 1/6) / (Gamma(1/3)^(n + 1/3) * exp(3*n^2/4 + n/3 + Pi/(18*sqrt(3)) - PolyGamma(1, 1/3) / (12*sqrt(3)*Pi) + 1/36)), where A = A074962 is the Glaisher-Kinkelin constant and PolyGamma(1, 1/3) = 10.09559712542709408179200409989... (PolyGamma[1, 1/3] in Mathematica or Psi(1, 1/3) in Maple).
PolyGamma(1, 1/3) = 3^(3/2) * A261024 + 2*Pi^2/3.
From Vladimir Reshetnikov, Nov 11 2015: (Start)
a(n) = 3^(n*(n+1)/2) * G(n+4/3) / (G(1/3) * Gamma(1/3)^(n+1)), where G(x) is the Barnes G-function.
a(n) ~ 3^(n*(n+1)/2) * exp(-(9*n^2+4*n-1)/12) * n^((18*n^2+12*n-1)/36) * (2*Pi)^((3*n+1)/6) / (A * G(1/3) * Gamma(1/3)^(n+1)).
Note that G(1/3) = 3^(1/72) * exp(1/9 + Pi/(18*sqrt(3)) - PolyGamma(1, 1/3)/(12*sqrt(3)*Pi)) / (A^(4/3) * Gamma(1/3)^(2/3)).
(End)

A263352 Expansion of Product_{k>=1} 1/(1 - x^(2*k+3))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 0, 3, 1, 4, 2, 5, 6, 7, 10, 9, 19, 14, 29, 23, 46, 38, 66, 64, 99, 107, 143, 171, 211, 270, 311, 418, 465, 633, 698, 945, 1049, 1399, 1579, 2052, 2364, 2997, 3527, 4366, 5219, 6339, 7686, 9197, 11234, 13321, 16340, 19261, 23622, 27796

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

From Vaclav Kotesovec, Oct 17 2015: (Start)
In general, if g.f. = Product_{k>=1} 1/(1-x^(2*k+v))^k and v>0 is odd, then a(n) ~ d2(v) * (2*n)^(v^2/24 - 25/36) * exp(-Pi^4 * v^2 / (1728*Zeta(3)) - Pi^2 * v * n^(1/3) /(3 * 2^(8/3) * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) / (sqrt(3*Pi) * Zeta(3)^(v^2/24 - 7/36)), where Zeta(3) = A002117.
d2(v) = exp(Integral_{x=0..infinity} 1/(x*exp((v-2)*x) * (exp(2*x)- 1)^2) - (3*v^2-2)/(24*x*exp(x)) + v/(4*x^2) - 1/(4*x^3) dx).
d2(v) = 2^(v/4 - 1/12) * exp(-Zeta'(-1)/2) / Product_{j=1..(v-1)/2} (2*j-1)!!, where Zeta'(-1) = A084448 and Product_{j=1..(v-1)/2} (2*j-1)!! = A057863((v-1)/2).
d2(v) = 2^(1/12 + v/4 - v^2/8) * exp(1/12) * Pi^(v/4) / (A * G(v/2 + 1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Eric Weisstein's World of Mathematics, Barnes G-Function.

Cf. A052847, A000219, A263150, A035528, A263395, A263396, A263397.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(d>1 and d::odd, (d-3)/2, 0),
      d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 17 2015

nmax = 60; CoefficientList[Series[Product[1/(1 - x^(2*k+3))^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 2^(25/72) * sqrt(A) * exp(-1/24 + 3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3) - Pi^4/(192*Zeta(3)) - Pi^2 * n^(1/3)/(2^(8/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * Zeta(3)^(13/72) * n^(23/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A263397 Expansion of Product_{k>=1} 1/(1 - x^(2*k+9))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 1, 7, 2, 8, 6, 9, 10, 10, 19, 11, 28, 13, 44, 15, 60, 20, 85, 29, 110, 44, 146, 69, 183, 111, 233, 171, 286, 262, 358, 391, 441, 568, 553, 808, 697, 1129, 898, 1543, 1174, 2080, 1563, 2766

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

In general, if g.f. = Product_{k>=1} 1/(1-x^(2*k+v))^k and v>0 is odd, then a(n) ~ d2(v) * (2*n)^(v^2/24 - 25/36) * exp(-Pi^4 * v^2 / (1728*Zeta(3)) - Pi^2 * v * n^(1/3) /(3 * 2^(8/3) * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) / (sqrt(3*Pi) * Zeta(3)^(v^2/24 - 7/36)), where Zeta(3) = A002117.
d2(v) = exp(Integral_{x=0..infinity} 1/(x*exp((v-2)*x) * (exp(2*x)- 1)^2) - (3*v^2-2)/(24*x*exp(x)) + v/(4*x^2) - 1/(4*x^3) dx).
d2(v) = 2^(v/4 - 1/12) * exp(-Zeta'(-1)/2) / Product_{j=1..(v-1)/2} (2*j-1)!!, where Zeta'(-1) = A084448 and Product_{j=1..(v-1)/2} (2*j-1)!! = A057863((v-1)/2).
d2(v) = 2^(1/12 + v/4 - v^2/8) * exp(1/12) * Pi^(v/4) / (A * G(v/2 + 1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Eric Weisstein's World of Mathematics, Barnes G-Function.

Cf. A035528 (v=-1), A263150 (v=1), A263352 (v=3), A263395 (v=5), A263396 (v=7).

with(numtheory):
a:= proc(n) option remember; local r; `if`(n=0, 1,
       add(add(`if`(irem(d-8, 2, 'r')=1, d*r, 0)
       , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..65);  # Alois P. Heinz, Oct 17 2015

nmax = 60; CoefficientList[Series[Product[1/(1 - x^(2*k+9))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(11*k)/(k*(1-x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^(11*k)/(k*(1-x^(2*k))^2)).

a(n) ~ 16 * 2^(61/72) * exp(-1/24 - 3*Pi^4/(64*Zeta(3)) - 3*Pi^2 * n^(1/3) / (2^(8/3) * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) * sqrt(A) * n^(193/72) / (4725*sqrt(3*Pi) * Zeta(3)^(229/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A121835 Hankel transform of A014307.

Original entry on oeis.org

1, 1, 6, 540, 1360800, 154314720000, 1154953090368000000, 786615912769073587200000000, 64289841661815089567953305600000000000, 803921077736369993656026249310037606400000000000000

Offset: 0

Philippe Deléham, Aug 28 2006

nonn

Georg Fischer, Table of n, a(n) for n = 0..24

Cf. A014307.

Cf. A000178, A057863.

a(n) = A000178(n)*A057863(n).

a(7), a(8) corrected and more terms from Georg Fischer, May 10 2024

A137592 Duplicate of A098694.

Original entry on oeis.org

1, 2, 48, 34560, 1393459200, 5056584744960000, 2422112183371431936000000, 211155601241022491077587763200000000, 4417964278440225627098723475313498521600000000000

Offset: 0

Paul Barry, Jan 28 2008

dead

Hankel transform of A112934(n+1). [From Paul Barry, Dec 04 2009]

a(n)=Product{k=0..n, (2(k+1)(2k+1))^(n-k)}; a(n)=A000178(n)*A057863(n)*A006125(n+1)=A121835(n)*A006125(n+1);

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 1/(1 + 1/(2*k+2)!/x/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013

cp's OEIS Frontend

A279886 a(n) = A057863(n+1) * Sum_{k=0..n}(k! / (2*k+1)!!).

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A005118 Number of simple allowable sequences on 1..n containing the permutation 12...n.

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A098694 Double-superfactorials: a(n) = Product_{k=1..n} (2k)!.

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A092042 Decimal expansion of e^(1/4).

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A086205 Determinant of n X n matrix M_(i,j) = binomial(i^2, j).

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A263416 a(n) = Product_{k=0..n} (3*k+1)^(n-k).

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A263352 Expansion of Product_{k>=1} 1/(1 - x^(2*k+3))^k.

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