A348607 -id:A348607 - cp's OEIS frontend

A010790 a(n) = n!*(n+1)!.

1, 2, 12, 144, 2880, 86400, 3628800, 203212800, 14631321600, 1316818944000, 144850083840000, 19120211066880000, 2982752926433280000, 542861032610856960000, 114000816848279961600000, 27360196043587190784000000, 7441973323855715893248000000

Offset: 0

N. J. A. Sloane

Let M_n be the symmetrical n X n matrix M_n(i,j)=1/min(i,j); then for n>=0 det(M_n)=(-1)^(n-1)/a(n-1). - Benoit Cloitre, Apr 27 2002
If n women and n men are to be seated around a circular table, with no two of the same sex seated next to each other, the number of possible arrangements is a(n-1). - Ross La Haye, Jan 06 2009
a(n-1) is also the number of (directed) Hamiltonian cycles in the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Jul 15 2011
a(n) is also number of ways to place k nonattacking semi-bishops on an n X n board, sum over all k>=0 (for definition see A187235). - Vaclav Kotesovec, Dec 06 2011
a(n) is number of permutations of {1,2,3,...,2n} such that no odd numbers are adjacent. - Ran Pan, May 23 2015
a(n) is number of permutations of {1,2,3,...,2n+1} such that no odd numbers are adjacent. - Ran Pan, May 23 2015
a(n-1) is the number of elements of the wreath product of S_n and S_2 with cycle partition equal to (2n), where S_n is the symmetric group of order n. - Josaphat Baolahy, Mar 12 2024

			G.f. = 1 + 2*x + 12*x^2 + 144*x^3 + 2880*x^4 + 86400*x^5 + ...

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, pp. 63-65.
Kenneth H. Rosen, Editor-in-Chief, Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000, page 91. [Ross La Haye, Jan 06 2009]

T. D. Noe, Table of n, a(n) for n = 0..100
J. Agapito, On symmetric polynomials with only real zeros and nonnegative gamma-vectors, Linear Algebra and its Applications, Volume 451, 15 June 2014, Pages 260-289.
Steve Gadbois, 104.12 From calendar coincidence to factorials to Ramanujan, The Mathematical Gazette (2020) Vol. 104, Issue 560, 304-306.
Anatol N. Kirillov, On some quadratic algebras. I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 002, 172 p. (2016).
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 268.
S. Tanimoto, Parity alternating permutations and signed Eulerian numbers, Ann. Comb. 14 (2010) 355 (total number of PAPs of [2n+1].)
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Shawn L. Witte, Link Nomenclature, Random Grid Diagrams, and Markov Chain Methods in Knot Theory, Ph. D. Dissertation, University of California-Davis (2020).
Index entries for sequences related to factorial numbers

Second column of triangle A129065.

Cf. A004737, A000290, A000108, A126120.

[Factorial(n)*Factorial(n+1): n in [0..20]]; // Vincenzo Librandi, Aug 08 2014

Maple
```
f:= n-> n!*(n+1)!: seq(f(n), n=0..30);
```

s=1;lst={s};Do[s+=(s*=n)*n;AppendTo[lst, s], {n, 1, 4!, 1}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
Times@@@Partition[Range[0,25]!,2,1] (* Harvey P. Dale, Jun 17 2011 *)

a(n)= n!^2*(n+1) \\ Charles R Greathouse IV, Jul 31 2011

from math import factorial
def A010790(n): return factorial(n)**2*(n+1) # Chai Wah Wu, Apr 22 2024

[stirling_number1(n,1)*factorial (n-2) for n in range(2, 17)] # Zerinvary Lajos, Jul 07 2009

From Karol A. Penson, Oct 23 2001: (Start)
Integral representation as n-th moment of a positive function f on the positive half axis, where f(x) = 2*sqrt(x)*BesselK(1, 2*sqrt(x)). Then:
a(n) = Integral_{x>=0} x^n * f(x) dx.
G.f.: a(0) = 1 and a(n) = subs(x=0, n!*diff(1/((x-1)^2), x$n)) for n >= 1. (End)
Sum_{i >=0} 1/a(i) = A096789. - Gerald McGarvey, Jun 10 2004
With b(n)=A002378(n) for n>0 and b(0)=1, a(n) = b(n)*b(n-1)...*b(0). - Tom Copeland, Sep 21 2011
a(n) = det(PS(i+1,j), 1 <= i,j <= n), where PS(n,k) are Legendre-Stirling numbers of the second kind. - Mircea Merca, Apr 04 2013
a(n) = (2*n)! / A000108(n) which implies that the e.g.f. of A126120 is Sum_{k>=0} x^(2*k) / a(k). - Michael Somos, Nov 15 2014
0 = a(n)*(+18*a(n+2) - 15*a(n+3) + a(n+4)) + a(n+1)*(-9*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2)) for all n>=0. - Michael Somos, Nov 15 2014
From Ilya Gutkovskiy, Jan 20 2017: (Start)
a(n) ~ 2*Pi*n^(2*n+2)/exp(2*n).
Sum_{n>=0} (-1)^n/a(n) = BesselJ(1,2) = 0.576724807756873387202448... = A348607 (End)
D-finite with recurrence: a(n) -n*(n+1)*a(n-1)=0. - R. J. Mathar, Jan 27 2020
a(n) = 1/([x^n] hypergeom([], [2], x)). - Peter Luschny, Sep 13 2024

A096789 Decimal expansion of BesselI(1,2).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 09 2004

			1.59063685463732906338225...

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products (6th ed.), 2000, (eq. 0.246.2).
Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind.

Cf. A000108, A070910, A228229, A348607.

evalf(BesselI(1,2)). # R. J. Mathar, Oct 16 2015

RealDigits[BesselI[1, 2], 10, 110][[1]]
(* Or *) RealDigits[ Sum[ n/(n!n!), {n, 0, Infinity}], 10, 110][[1]]

besseli(1,2) \\ Charles R Greathouse IV, Feb 19 2014

Equals Sum_{k >= 0} k/k!^2.
Continued fraction expansion: 1/(1 - 1/(3 - 2/(7 - 6/(13 - 12/(21 - ... - n*(n-1)/(n^2+n+1 - ...)))))). For a sketch of the proof see A228229. Cf. A070910. - Peter Bala, Aug 19 2013
From Amiram Eldar, Jul 09 2023: (Start)
Equals exp(-2) * Sum_{k>=1} A000108(k)/(k-1)!.
Equals exp(2) * Sum_{k>=1} (-1)^(k+1) * A000108(k)/(k-1)!. (End)

A301484 Decimal expansion of J_0(2)/J_1(2) = 1 - 1/(2 - 1/(3 - 1/(4 - ...))).

Original entry on oeis.org

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

Offset: 0

David H Bailey, Mar 22 2018

These are the first 105 decimal digits of the constant defined by the continued fraction 1 - 1/(2 - 1/(3 - 1/(4 - ... -1/m))) as m goes to infinity. The continued fraction appears to converge fairly rapidly. Just 50 terms, for instance, suffices to produce a numerical value that appears to be good to 100 digits, based on comparisons with more terms and higher precision. This sequence was brought to the author's attention by Beresford Parlett of U.C. Berkeley.
Addendum: This sequence has been identified by Karl Dilcher. He noted that the sequence of continued fraction convergents is the same as A058797. In short, the real constant whose decimal expansion is given above is given by BesselJ[0,2]/BesselJ[1,2] (Mathematica expression). The comments at A058797 have quite a bit of additional information and references.
Equivalent to the imaginary part of the infinite continued fraction i + 1/(2i + 1/(3i + ...)) where i = sqrt(-1). - Matthew Niemiro, Dec 22 2019

			0.38821076556779578751165855730653702922174504072532981864642827593735174...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A058797, A091681, A296168.

SetDefaultRealField(RealField(100)); BesselFunction(0, 2)/BesselFunction(1, 2); // G. C. Greubel, Dec 31 2019

evalf(BesselJ(0, 2)/BesselJ(1, 2), 100); # G. C. Greubel, Dec 31 2019

1 +ContinuedFractionK[(-1)^(n+1)*n, {n,2,Infinity}]
N[1+ContinuedFractionK[(-1)^(n+1)*n, {n,2,50}], 105] (* 105 decimals *)
RealDigits[BesselJ[0, 2]/BesselJ[1, 2], 10, 100][[1]] (* G. C. Greubel, Dec 31 2019 *)

default(realprecision, 100); besselj(0,2)/besselj(1,2) \\ Altug Alkan, Mar 22 2018

numerical_approx(bessel_J(0,2)/bessel_J(1,2), digits=100) # G. C. Greubel, Dec 31 2019

Equals BesselJ(0,2)/BesselJ(1,2) = A091681 / A348607.

cp's OEIS Frontend

A010790 a(n) = n!*(n+1)!.

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A096789 Decimal expansion of BesselI(1,2).

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A301484 Decimal expansion of J_0(2)/J_1(2) = 1 - 1/(2 - 1/(3 - 1/(4 - ...))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula