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%I A001813 M2040 N0808
%S A001813 1,2,12,120,1680,30240,665280,17297280,518918400,17643225600,
%T A001813 670442572800,28158588057600,1295295050649600,64764752532480000,
%U A001813 3497296636753920000,202843204931727360000,12576278705767096320000
%N A001813 Quadruple factorial numbers: (2n)!/n!.
%C A001813 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.
%C A001813 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
%C A001813 a(n)=12*A051618(a) n>=2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Feb 15 2008
%C A001813 Hankel transform is A137565. [From Paul Barry (pbarry(AT)wit.ie), Nov 
               25 2009]
%D A001813 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001813 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001813 P. Cvitanovic, Group theory for Feynman diagrams in non-Abelian gauge 
               theories, Phys. Rev. D 14 (1976), 1536-1553.
%D A001813 D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.6, Eq. 32.
%D A001813 L. C. Larson, The number of essentially different nonattacking rook arrangements, 
               J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
%D A001813 McDonnell, Eugene, "Magic Squares and Permutations" APL Quote-Quad 7.3 
               (Fall, 1976)
%D A001813 R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial 
               Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
%D A001813 H. E. Salzer, Coefficients for expressing the first thirty powers in 
               terms of the Hermite polynomials, Math. Comp., 3 (1948), 167-169.
%H A001813 N. J. A. Sloane, <a href="b001813.txt">Table of n, a(n) for n = 0..100</
               a>
%H A001813 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A001813 W. Y. C. Chen, L. W. Shapiro and L. L. M. Young, <a href="http://arXiv.org/
               abs/math.CO/0503300">Parity reversing involutions on plane trees 
               and 2-Motzkin paths</a>
%H A001813 P. Cvitanovic, <a href="http://www.nbi.dk/~predrag/papers/PRD14-76.pdf">
               Group theory for Feynman diagrams in non-Abelian gauge theories</
               a>, Phys. Rev. D14 (1976), 1536-1553.
%H A001813 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=115">
               Encyclopedia of Combinatorial Structures 115</a>
%H A001813 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">
               Sobalian Coefficients</a>.
%H A001813 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/
               index.html">Miscellaneous</a>.
%H A001813 E. Lucas, <a href="http://gallica.bnf.fr/scripts/ConsultationTout.exe?E=0&O=N029021">
               Th\'{e}orie des Nombres</a>. Gauthier-Villars, Paris, 1891, Vol. 
               1, p. 221.
%H A001813 R. J. Marsh and P. P. Martin, <a href="http://arXiv.org/abs/math.CO/0612572">
               Pascal arrays: counting Catalan sets</a>
%H A001813 C. Radoux, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28radoux.html">
               Determinants de Hankel et theoreme de Sylvester</a>
%H A001813 <a href="Sindx_Par.html#partN">Index entries for related partition-counting 
               sequences</a>
%H A001813 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               127
%F A001813 E.g.f. (1-4*x)^(-1/2). a(n) = (2*n)!/n! = product[ k=0..n-1 ] (4*k+2).
%F A001813 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 (penson(AT)lptl.jussieu.fr), Sep 18 2001
%F A001813 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 (benoit7848c(AT)orange.fr), Apr 27 2003
%F A001813 With interpolated zeros (1, 0, 2, 0, 12, ...) this has e.g.f. exp(x^2). 
               - Paul Barry (pbarry(AT)wit.ie), May 09 2003
%F A001813 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
%F A001813 For asymptotics see the Robinson paper.
%F A001813 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
%F A001813 a(n)=A000984(n)*A000142(n) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 25 2008
%F A001813 Let b(n)=b(n-1)+4; then a(n)=b(n)*a(n-1). - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), 
               Sep 17 2008
%F A001813 Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 15 2009: (Start)
%F A001813 G.f.: 1/(1-2x/(1-4x/(1-6x/(1-8x/(1-10x/(1-...... (continued fraction);
%F A001813 a(n)=(n+1)!*A000108(n); (End)
%F A001813 a(n)=sum{k=0..n, A132393(n,k)*2^(2n-k)}. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Feb 10 2009]
%F A001813 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 (pbarry(AT)wit.ie), 
               Nov 25 2009]
%e A001813 The following permutations of order 8 and their reversals have this property:
%e A001813 1 7 3 5 2 4 0 6
%e A001813 1 7 4 2 5 3 0 6
%e A001813 2 3 7 6 1 0 4 5
%e A001813 2 4 7 1 6 0 3 5
%e A001813 3 2 6 7 0 1 5 4
%e A001813 3 5 1 7 0 6 2 4
%p A001813 A001813 := n->(2*n)!/n!;
%p A001813 spec := [ B, {B=Union(Z,Prod(B,B))}, labeled ]; [seq(combstruct[count](spec, 
               size=n), n=1..20)];
%p A001813 For Maple program see A000903.
%p A001813 BB:=[T,{T=Prod(Z,F),F=Sequence(B),B=Prod(F,Z)}, labeled]: seq(count(BB,
               size=i),i=1..17); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 22 2007
%p A001813 seq(mul((n+k), k=1..n), n=0..16); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Feb 15 2008
%t A001813 k = 4; b[1] = 2; b[n_] := b[n] = b[n - 1] + k; a[0] = 1; a[1] = 2; a[n_] 
               := a[n] = a[n - 1]*b[n]; Table[a[n], {n, 0, 20}] - Roger L. Bagula 
               (rlbagulatftn(AT)yahoo.com), Sep 17 2008
%t A001813 s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 1, 5!, 4}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Nov 08 2008]
%o A001813 (Mupad) combinat::catalan(n)*(n+1)*n! $ n = 0..16 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Feb 15 2007
%o A001813 (Other) sage: [binomial(2*n,n)*factorial(n) for n in xrange(0, 17)]# 
               [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 03 2009]
%Y A001813 Cf. A037224, A048854, A001147, A007696, A008545, A122670 (essentially 
               the same sequence).
%Y A001813 Cf. A000165, A047055, A047657, A084947, A084948, A084949.
%Y A001813 Cf. A010050, A000142, A008275, A000108, A000984, A008276, A000680, A094216.
%Y A001813 Sequence in context: A096317 A081470 A108135 this_sequence A097388 A152188 
               A131815
%Y A001813 Adjacent sequences: A001810 A001811 A001812 this_sequence A001814 A001815 
               A001816
%K A001813 nonn,easy,nice,new
%O A001813 0,2
%A A001813 N. J. A. Sloane (njas(AT)research.att.com).
%E A001813 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 01 2000

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