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A001813
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Quadruple factorial numbers: (2*n)!/n!.
(Formerly M2040 N0808)
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75
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1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400, 17643225600, 670442572800, 28158588057600, 1295295050649600, 64764752532480000, 3497296636753920000, 202843204931727360000, 12576278705767096320000
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OFFSET
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0,2
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COMMENTS
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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 (zerinvarylajos(AT)yahoo.com), 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. [From Paul Barry, Nov 25 2009]
For n>0: a(n) = A173333(2*n,n); cf. A006963, A001761. [From Reinhard Zumkeller, Feb 19 2010]
The e.g.f. of 1/a(n)=n!/(2*n)! is (exp(sqrt(x)) + exp(-sqrt(x)) )/2. [From Wolfdieter Lang, Jan 09 2012]
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REFERENCES
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P. Cvitanovic, Group theory for Feynman diagrams in non-Abelian gauge theories, Phys. Rev. D 14 (1976), 1536-1553.
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).
H. E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., 3 (1948), 167-169.
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).
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 0..100
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
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
C. Radoux, Determinants de Hankel et theoreme de Sylvester
Index to divisibility sequences
Index entries for related partition-counting sequences
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FORMULA
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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) - Andre F. Labossiere (boronali(AT)laposte.net), Jun 21 2007
a(n) = A000984(n)*A000142(n) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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
Contribution 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)}. [From Philippe DELEHAM, 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). [From 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. [From 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
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EXAMPLE
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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
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MAPLE
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A001813 := n->(2*n)!/n!;
A001813 := n -> mul(k, k = select(k-> k mod 4 = 2, [$1 .. 4*n])): seq(A001813(n), n=0..16);
- Peter Luschny, Jun 23 2011
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MATHEMATICA
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Table[(2n)!/n!, {n, 0, 20}] *( From Harvey P. Dale, May 02 2011 *)
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PROG
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(Sage) [binomial(2*n, n)*factorial(n) for n in xrange(0, 17)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 03 2009]
(PARI) a(n)=binomial(n+n, n)*n! \\ Charles R Greathouse IV, Jun 15 2011
(Maxima) A001813(n):=binomial(n+n, n)*n!$
makelist(A001813(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */
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CROSSREFS
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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.
Sequence in context: A081470 A108135 * A097388 A215188 A131815 A177774
Adjacent sequences: A001810 A001811 A001812 * A001814 A001815 A001816
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from James A. Sellers, May 01 2000
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STATUS
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approved
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