cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A001818 Squares of double factorials: (1*3*5*...*(2n-1))^2 = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 1, 9, 225, 11025, 893025, 108056025, 18261468225, 4108830350625, 1187451971330625, 428670161650355625, 189043541287806830625, 100004033341249813400625, 62502520838281133375390625, 45564337691106946230659765625, 38319607998220941779984862890625
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Number of permutations in S_{2n} in which all cycles have even length (cf. A087137).
Also number of permutations in S_{2n} in which all cycles have odd length. - Vladeta Jovovic, Aug 10 2007
a(n) is the sum over all multinomials M2(2*n,k), k from {1..p(2*n)} restricted to partitions with only even parts. p(2*n)= A000041(2*n) (partition numbers) and for the M2-multinomial numbers in A-St order see A036039(2*n,k). - Wolfdieter Lang, Aug 07 2007
From Zhi-Wei Sun, Jun 26 2022: (Start)
Conjecture 1: For any primitive 2n-th root zeta of unity, the permanent of the 2n X 2n matrix [m(j,k)]_{j,k=1..2n} coincides with a(n) = ((2n-1)!!)^2, where m(j,k) is (1+zeta^(j-k))/(1-zeta^(j-k)) if j is not equal to k, and 1 otherwise.
The determinant of [m(j,k)]_{j,k=1..2n} was shown to be (-1)^(n-1)*((2n-1)!!)^2/(2n-1) by Han Wang and Zhi-Wei Sun in 2022.
Conjecture 2: Let p be an odd prime. Then the permanent of (p-1) X (p-1) matrix [f(j,k)]_{j,k=1..p-1} is congruent to a((p-1)/2) = ((p-2)!!)^2 modulo p^2, where f(j,k) is (j+k)/(j-k) if j is not equal to k, and f(j,k) = 1 otherwise. (End)

Examples

			Multinomial representation for a(2): partitions of 2*2=4 with even parts only: (4) with position k=1, (2^2) with k=3; M2(4,1)= 6 and M2(4,3)= 3, adding up to a(2)=9.
G.f. = 1 + x + 9*x^2 + 225*x^3 + 11025*x^4 + 893025*x^5 + 108056025*x^6 + ...

References

  • John Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.34(c).

Links

Crossrefs

Cf. A001147, A002454, A111595, A197037.
Bisection of A012248.
Right-hand column 1 in triangle A008956.

Programs

  • Magma
    DoubleFactorial:=func< n | &*[n..2 by -2] >; [DoubleFactorial((2*n-1))^2: n in [0..20] ]; // Vincenzo Librandi, Jul 21 2017
  • Maple
    a := proc(m) local k; 4^m*mul((-1)^k*(k-m-1/2),k=1..2*m) end; # Peter Luschny, Jun 01 2009
  • Mathematica
    FoldList[Times,1,Range[1,25,2]]^2 (* or *) Join[{1},(Range[1,29,2]!!)^2] (* Harvey P. Dale, Jun 06 2011, Apr 10 2012 *)
Table[((2 n - 1)!!)^2, {n, 0, 30}] (* Vincenzo Librandi, Jul 21 2017 *)
  • PARI
    a(n)=((2*n)!/(n!*2^n))^2
  • PARI
    {a(n) = if( n<0, 1 / a(-n), sqr((2*n)! / (n! * 2^n)))}; /* Michael Somos, Jan 06 2017 */

Formula

a(n) = A001147(n)^2.
a(n) = A111595(2*n, 0).
a(n) = (2*n-1)!*Sum_{k=0..n-1} binomial(2*k,k)/4^k, n >= 1. - Wolfdieter Lang, Aug 23 2005
arcsinh(x) = Sum_{n>=1} (-1)^(n-1)*a(n)*x^(2*n-1)/(2*n-1)!. - James R. Buddenhagen, Mar 24 2009
From Karol A. Penson, Oct 21 2009: (Start)
G.f.: Sum_{n>=0} a(n)*x^n/(n!)^2 = 2*EllipticK(2*sqrt(x))/Pi.
Asymptotically: a(n) = (2/((exp(-1/2))^2*(exp(1/2))^2)-1/(6*(exp(-1/2))^2*(exp(1/2))^2*n)+1/(144*(exp(-1/2))^2*(exp(1/2))^2*n^2)+O(1/n^3))*(2^n)^2/(((1/n)^n)^2*(exp(n))^2), n->infinity.
Integral representation as n-th moment of a positive function on a positive halfaxis (solution of the Stieltjes moment problem), in Maple notation:
a(n) = Integral_{x>=0} x^n*BesselK(0,sqrt(x))/(Pi*sqrt(x)).
This solution is unique.
(End)
D-finite with recurrence: a(0) = 1, a(n) = (2*n-1)^2*a(n-1), n > 0.
a(n) ~ 2*2^(2*n)*e^(-2*n)*n^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
E.g.f.: 1/sqrt(1-x^2) = Sum_{n >= 0} a(n)*x^(2*n)/(2*n)!. Also arcsin(x) = Sum_{n >= 0} a(n)*x^(2*n+1)/(2*n+1)!. - Michael Somos, Jul 03 2002
(-1)^n*a(n) is the coefficient of x^0 in prod(k=1, 2*n, x+2*k-2*n-1). - Benoit Cloitre and Michael Somos, Nov 22 2002
-arccos(x) + Pi/2 = x + x^3/3! + 9*x^5/5! + 225*x^7/7! + 11205*x^9/9! + ... - Tom Copeland, Oct 23 2008
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (4*k^2+4*k+1)/(1-x/(x - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
a(n) = det(V(i+1,j), 1 <= i,j <= n), where V(n,k) are central factorial numbers of the second kind with odd indices. - Mircea Merca, Apr 04 2013
a(n) = (1+x^2)^(n+1/2) * (d/dx)^(2*n) (1+x^2)^(n-1/2). See Tao link. - Robert Israel, Jun 04 2015
a(n) = 4^n * gamma(n + 1/2)^2 / Pi. - Daniel Suteu, Jan 06 2017
0 = a(n)*(+384*a(n+2) - 60*a(n+3) + a(n+4)) + a(n+1)*(-36*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2)) and a(n) = 1/a(-n) for all n in Z. - Michael Somos, Jan 06 2017
From Robert FERREOL, Jul 30 2020: (Start)
a(n) = ((2*n)!/4^n)*binomial(2*n,n).
a(n) = (2*n-1)!*Sum_{k=0..n-1} a(k)/(2*k)!, n >= 1.
a(n) = A184877(2*n-1) for n>=1. (End)
From Amiram Eldar, Mar 18 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + L_0(1)*Pi/2, where L is the modified Struve function (see A197037).
Sum_{n>=0} (-1)^n/a(n) = 1 - H_0(1)*Pi/2, where H is the Struve function. (End)

Extensions

Incorrect formula deleted by N. J. A. Sloane, Jul 03 2009

A296466 Expansion of e.g.f. arcsinh(arcsin(x)) (odd powers only).

Original entry on oeis.org

1, 0, 8, 56, 8000, 342144, 68623488, 8295676416, 2411783847936, 584142614728704, 240810283258527744, 96772676958798741504, 54867909992513301282816, 32661008325245409302937600, 24691868812821871169667072000, 20243305132513358736699378892800, 19829947398943934886214249532620800
Offset: 0

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Author

Ilya Gutkovskiy, Dec 13 2017

Keywords

Examples

			arcsinh(arcsin(x)) = x/1! + 8*x^5/5! + 56*x^7/7! + 8000*x^9/9! + 342144*x^11/11! + 68623488*x^13/13! + ...

Crossrefs

Cf. A001818, A003722, A012248, A296464.

Programs

  • Mathematica
    nmax = 17; Table[(CoefficientList[Series[ArcSinh[ArcSin[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 17; Table[(CoefficientList[Series[Log[Sqrt[1 - Log[I x + Sqrt[1 - x^2]]^2] - I Log[I x + Sqrt[1 - x^2]]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

Formula

E.g.f.: arcsin(arcsinh(x)) (odd powers only, absolute values).
E.g.f.: log(sqrt(1 - log(i*x + sqrt(1 - x^2))^2) - i*log(i*x + sqrt(1 - x^2))), where i is the imaginary unit (odd powers only).
a(n) ~ 2 * (2*n)! / sqrt(Pi*(4 + Pi^2)*n). - Vaclav Kotesovec, Dec 13 2017

A012115 Expansion of e.g.f. of cos(arcsin(arcsinh(x))), even powers only.

Original entry on oeis.org

1, -1, 1, -49, 897, -96801, 6803457, -1275363153, 207592347393, -60889593787713, 17888210916886017, -7679611833095218545, 3530100224793651058305, -2109645360096014314212705, 1387893136694794953345211905, -1109191050707015380265790041745, 984063632341385922413073350925825
Offset: 0

Views

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Keywords

Examples

			1 - 1/2!*x^2 + 1/4!*x^4 - 49/6!*x^6 + 897/8!*x^8...

Links

Crossrefs

Cf. A012248.

Programs

  • Mathematica
    With[{nn = 50}, Table[(CoefficientList[Series[Cos[ArcSin[ArcSinh[x]]], {x, 0, 2*nn}], x]*Range[0, 2*nn]!)[[n]], {n, 1, 2*nn + 1, 2}]] (* G. C. Greubel, Apr 12 2017 *)
  • PARI
    x = 'x + O('x^40); select(x->x, Vec(serlaplace(cos(asin(asinh(x)))))) \\ Michel Marcus, Mar 09 2017

Extensions

More terms from Michel Marcus, Mar 09 2017
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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