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A001813 Quadruple factorial numbers: (2*n)!/n!.
(Formerly M2040 N0808)
76
1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400, 17643225600, 670442572800, 28158588057600, 1295295050649600, 64764752532480000, 3497296636753920000, 202843204931727360000, 12576278705767096320000 (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

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

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

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

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)

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)

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

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

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.

W. Y. C. Chen, L. W. Shapiro and L. L. M. Young, Parity reversing involutions on plane trees and 2-Motzkin paths

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

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, 2014

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

E. Lucas, Th\'{e}orie 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.

R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv math.CO.0606404.

C. Radoux, Determinants de Hankel et theoreme de Sylvester

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(4*i+2, i=0..n-1) = 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) = 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) = 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, ...

...

- Gary W. Adamson, Jul 19 2011

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

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

(Maxima) A001813(n):=binomial(n+n, n)*n!$

makelist(A001813(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */

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.

Sequence in context: A081470 A108135 * A097388 A215188 A236357 A226759

Adjacent sequences:  A001810 A001811 A001812 * A001814 A001815 A001816

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers, May 01 2000

STATUS

approved

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Last modified December 22 21:10 EST 2014. Contains 252372 sequences.