login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation to keep the OEIS running. In 2018 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001813 Quadruple factorial numbers: a(n) = (2n)!/n!.
(Formerly M2040 N0808)
118
1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400, 17643225600, 670442572800, 28158588057600, 1295295050649600, 64764752532480000, 3497296636753920000, 202843204931727360000, 12576278705767096320000, 830034394580628357120000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Counts binary rooted trees (with out-degree <= 2), embedded in plane, with n labeled end nodes of degree 1. Unlabeled version gives Catalan numbers A000108.

Define a "downgrade" to be the permutation which places the items of a permutation in descending order. We are concerned with permutations that are identical to their downgrades. Only permutations of order 4n and 4n+1 can have this property; the number of permutations of length 4n having this property are equinumerous with those of length 4n+1. If a permutation p has this property then the reversal of this permutation also has it. a(n) = number of permutations of length 4n and 4n+1 that are identical to their downgrades. - Eugene McDonnell (eemcd(AT)mac.com), Oct 26 2003

Number of broadcast schemes in the complete graph on n+1 vertices, K_{n+1}. - Calin D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008

Hankel transform is A137565. - Paul Barry, Nov 25 2009

The e.g.f. of 1/a(n) = n!/(2*n)! is (exp(sqrt(x)) + exp(-sqrt(x)) )/2. - Wolfdieter Lang, Jan 09 2012

From Tom Copeland, Nov 15 2014: (Start)

Aerated with intervening zeros (1,0,2,0,12,0,120,...)=a(n) (cf. A123023 and A001147), the e.g.f. is e^(t^2), so this is the base for the Appell sequence with e.g.f. e^(t^2) e^(x*t) = exp(P(.,x),t) (reverse A059344, cf. A099174, A066325 also). P(n,x) = (a. + x)^n with (a.)^n = a_n and comprise the umbral compositional inverses for e^(-t^2)e^(x*t) = exp(UP(.,x),t), i.e., UP(n,P(.,t))= x^n = P(n,UP(.,t)), e.g., (P(.,t))^n = P(n,t).

Equals A000407*2 with leading 1 added. (End)

a(n) is also the number of square roots of any permutation in S_{4*n} whose disjoint cycle decomposition consists of 2*n transpositions. - Luis Manuel Rivera Martínez, Mar 04 2015

Self-convolution gives A076729. - Vladimir Reshetnikov, Oct 11 2016

For n > 1, it follows from the formula dated Aug 07 2013 that a(n) is a Zumkeller number (A083207). - Ivan N. Ianakiev, Feb 28 2017

For n divisible by 4, a(n/4) is the number of ways to place n points on an n X n grid with pairwise distinct abscissae, pairwise distinct ordinates, and 90-degree rotational symmetry. For n == 1 (mod 4), the number of ways is a((n-1)/4) because the center point can be considered "fixed". For 180-degree rotational symmetry see A006882, for mirror symmetry see A000085, A135401, and A297708. - Manfred Scheucher, Dec 29 2017

REFERENCES

D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.6, Eq. 32.

L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.

McDonnell, Eugene, "Magic Squares and Permutations" APL Quote-Quad 7.3 (Fall, 1976)

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).

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).

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..100

Murray Bremner, Martin Markl, Distributive laws between the Three Graces, arXiv:1809.08191 [math.AT], 2018.

R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015) # 15.3.2.

Peter J. Cameron, Some treelike objects, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See p. 155. - N. J. A. Sloane, Apr 18 2014

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Elliot J. Carr, Matthew J. Simpson, New homogenization approaches for stochastic transport through heterogeneous media, arXiv:1810.08890 [physics.bio-ph], 2018.

W. Y. C. Chen, L. W. Shapiro and L. L. M. Young, Parity reversing involutions on plane trees and 2-Motzkin paths, arXiv:math/0503300 [math.CO], 2005.

P. Cvitanovic, Group theory for Feynman diagrams in non-Abelian gauge theories, Phys. Rev. D14 (1976), 1536-1553.

Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018.

John Engbers, David Galvin, Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016. See p. 8.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 127

S. Goodenough, C. Lavault, On subsets of Riordan subgroups and Heisenberg--Weyl algebra, arXiv preprint arXiv:1404.1894 [cs.DM], 2014.

S. Goodenough, C. Lavault, Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups, The Electronic Journal of Combinatorics, 22(4) (2015), #P4.16,

H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy)

A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 115

L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]

Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin. 52 (2012), 41-54 (Theorem 1).

Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.

E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.

R. J. Marsh and P. P. Martin, Pascal arrays: counting Catalan sets, arXiv:math/0612572 [math.CO], 2006.

Calin D. Morosan, On the number of broadcast schemes in networks, Information Processing Letters, Volume 100, Issue 5 (2006), 188-193.

R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.

C. Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.

J. Riordan, Letter to N. J. A. Sloane, Feb 03 1975 (with notes by njas)

H. E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., 3 (1948), 167-169.

Index to divisibility sequences

Index entries for related partition-counting sequences

FORMULA

E.g.f.: (1-4*x)^(-1/2).

a(n) = (2*n)!/n! = Product_{k=0..n-1} (4*k + 2).

Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n) = int(x^n*exp(-x/4)/(sqrt(x)*2*sqrt(Pi)), x=0..infinity), n=0, 1, .. . This representation is unique. - Karol A. Penson, Sep 18 2001

Define a'(1)=1, a'(n) = Sum_{k=1..n-1} a'(n-k)*a'(k)*C(n, k); then a(n)=a'(n+1). - Benoit Cloitre, Apr 27 2003

With interpolated zeros (1, 0, 2, 0, 12, ...) this has e.g.f. exp(x^2). - Paul Barry, May 09 2003

a(n) = A000680(n)/A000142(n)*A000079(n) = Product_{i=0..n-1} (4*i + 2) = 4^n*Pochhammer(1/2, n) = 4^n*GAMMA(n+1/2)/sqrt(Pi). - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

For asymptotics, see the Robinson paper.

a(k) = (2*k)!/k! = Sum_{i=1..k+1} |A008275(i,k+1)| * k^(i-1). - André F. Labossière, Jun 21 2007

a(n) = 12*A051618(a) n >= 2. - Zerinvary Lajos, Feb 15 2008

a(n) = A000984(n)*A000142(n). - Zerinvary Lajos, Mar 25 2008

a(n) = A016825(n-1)*a(n-1). - Roger L. Bagula, Sep 17 2008

a(n) = (-1)^n*A097388(n). - D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008

From Paul Barry, Jan 15 2009: (Start)

G.f.: 1/(1-2x/(1-4x/(1-6x/(1-8x/(1-10x/(1-... (continued fraction);

a(n) = (n+1)!*A000108(n). (End)

a(n) = Sum_{k=0..n} A132393(n,k)*2^(2n-k). - Philippe Deléham, Feb 10 2009

G.f.: 1/(1-2x-8x^2/(1-10x-48x^2/(1-18x-120x^2/(1-26x-224x^2/(1-34x-360x^2/(1-42x-528x^2/(1-... (continued fraction). - Paul Barry, Nov 25 2009

a(n) = A173333(2*n,n) for n>0; cf. A006963, A001761. - Reinhard Zumkeller, Feb 19 2010

From Gary W. Adamson, Jul 19 2011: (Start)

a(n) = upper left term of M^n, M = an infinite square production matrix as follows:

  2, 2, 0, 0, 0, 0, ...

  4, 4, 4, 0, 0, 0, ...

  6, 6, 6, 6, 0, 0, ...

  8, 8, 8, 8, 8, 0, ...

  ...

(End)

G.f.: If G_N(x)=1+sum('(2*k)!*(x^k)/k!', 'k'=1..N),  G_N(x)=1+2*x/(G(0)-2*x); G(k)=1+2*x+4*x*k-2*x*(2*k+3)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011

a(n) = (-2)^n*Sum_{k=0..n} 2^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012

G.f.: 1/Q(0), where Q(k)= 1 + x*(4*k+2) - x*(4*k+4)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 18 2013

G.f.: 2/G(0), where G(k)= 1 + 1/(1 - x*(8*k+4)/(x*(8*k+4) - 1 + 8*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x/(2*x + 1/(2*k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013

a(n) = (4n-6)*a(n-2) + (4n-3)*a(n-1), n>=2. - Ivan N. Ianakiev, Aug 07 2013

Sum_{n>=0} 1/a(n) = (exp(1/4)*sqrt(Pi)*erf(1/2) + 2)/2 = 1 + A214869, where erf(x) is the error function. - Ilya Gutkovskiy, Nov 10 2016

EXAMPLE

The following permutations of order 8 and their reversals have this property:

1 7 3 5 2 4 0 6

1 7 4 2 5 3 0 6

2 3 7 6 1 0 4 5

2 4 7 1 6 0 3 5

3 2 6 7 0 1 5 4

3 5 1 7 0 6 2 4

MAPLE

A001813 := n->(2*n)!/n!;

A001813 := n -> mul(k, k = select(k-> k mod 4 = 2, [$1 .. 4*n])): seq(A001813(n), n=0..16);

# Peter Luschny, Jun 23 2011

MATHEMATICA

Table[(2n)!/n!, {n, 0, 20}] (* Harvey P. Dale, May 02 2011 *)

PROG

(Sage) [binomial(2*n, n)*factorial(n) for n in xrange(0, 17)] # Zerinvary Lajos, Dec 03 2009

(PARI) a(n)=binomial(n+n, n)*n! \\ Charles R Greathouse IV, Jun 15 2011

(PARI) first(n) = x='x+O('x^n); Vec(serlaplace((1 - 4*x)^(-1/2))) \\ Iain Fox, Jan 01 2018 (corrected by Iain Fox, Jan 11 2018)

(Maxima) makelist(binomial(n+n, n)*n!, n, 0, 30); /* Martin Ettl, Nov 05 2012 */

(MAGMA) [Factorial(2*n)/Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 09 2018

(GAP) List([0..20], n->Factorial(2*n)/Factorial(n)); # Muniru A Asiru, Nov 01 2018

CROSSREFS

Cf. A037224, A048854, A001147, A007696, A008545, A122670 (essentially the same sequence), A000165, A047055, A047657, A084947, A084948, A084949, A010050, A000142, A008275, A000108, A000984, A008276, A000680, A094216.

Catalan(n-1)*M^(n-1)*n! for M=1,2,3,4,5,6: A001813, A052714 (or A144828), A221954, A052734, A221953, A221955.

Cf. A123023, A001147, A059344, A099174, A066325, A001700, A000407, A006882.

Cf. A000085, A006882, A135401, A297708. - Manfred Scheucher, Jan 07 2018

Sequence in context: A081470 A108135 A097388 * A215188 A236357 A226759

Adjacent sequences:  A001810 A001811 A001812 * A001814 A001815 A001816

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers, May 01 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 16 04:04 EST 2018. Contains 318158 sequences. (Running on oeis4.)