Search: id:A003149
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%I A003149 M1496
%S A003149 1,2,5,16,64,312,1812,12288,95616,840960,8254080,89441280,1060369920,
%T A003149 13649610240,189550368000,2824077312000,44927447040000,760034451456000,
%U A003149 13622700994560000,257872110354432000,5140559166898176000
%N A003149 Sum_{k=0..n} k!(n-k)!.
%C A003149 The sequence (origin 1) is the resistance between opposite corners of 
               an n-dimensional hypercube of unit resistors, multiplied by n!.
%C A003149 The resistances for n = 1,2,3,... are 1 1 5/6 2/3 8/15 13/30 151/420 
               32/105 83/315 73/315 1433/6930 ...
%C A003149 Number of {12,21*,2*1}-avoiding signed permutations in the hyperoctahedral 
               group.
%C A003149 a(n) is the sum of the reciprocals of the binomial coefficients C(n,k), 
               multiplied by n!; example : a(4) = 4!*(1/1 + 1/4 + 1/6 + 1/4 + 1/
               1) = 64 . - Philippe DELEHAM, May 12 2005
%C A003149 a(n) = number of permutations on [n+1] that avoid the pattern 13-2|. 
               The absence of a dash between 1 and 3 means the "1" and "3" must 
               be consecutive in the permutation; the vertical bar means the "2" 
               must occur at the end of the permutation. For example, 24153 fails 
               to avoid this pattern: 243 is an offending subpermutation. - David 
               Callan (callan(AT)stat.wisc.edu), Nov 02 2005
%C A003149 n!/A003149(n) is the probability that a random walk on an (n+1)-dimensional 
               hypercube will visit the diagonally opposite vertex before it returns 
               to its starting point. 2^n*A003149(n)/n! is the expected length of 
               a random walk from one vertex of an (n+1)-dimensional hypercube to 
               the diagonally opposite vertex (a walk which may include one or more 
               passes through the starting point). These "random walk" examples 
               are solutions to IBM's "Ponder This" puzzle for April, 2006 - Graeme 
               McRae (g_m(AT)mcraefamily.com), Apr 02 2006
%C A003149 a(n) = number of strong fixed points in all permutations of {1,2,...,
               n+1} (a permutation p of {1,2,...,n} is said to have j as a strong 
               fixed point (splitter) if p(k)<j for k<j and p(k)>j for k>j). Example: 
               a(2)=5 because the permutations of {1,2,3}, with marked strong fixed 
               points, are: 1'2'3', 1'32, 312, 213', 231 and 321. [From Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Oct 28 2008]
%D A003149 Author?, Resistances in the multidimensional cube, Quantum 7:1 (1996) 
               12-15 and 63.
%D A003149 Problem E3467, Amer. Math. Monthly, 100 (1993), 800-801. [From Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), Oct 28 2008]
%D A003149 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003149 Stanley, R. P., Enumerative Combinatorics, Volume 1 (1986), p. 49. [From 
               Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 28 2008]
%D A003149 B. Sury, Sum of the reciprocals of the binomial coefficients, Europ. 
               J. Comb., 14 (1993), 351-353.
%H A003149 T. D. Noe, <a href="b003149.txt">Table of n, a(n) for n=0..100</a>
%H A003149 Fred Curtis, <a href="http://f2.org/maths/resnet/">Resistance-network 
               Problems</a>.
%H A003149 IBM, <a href="http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/challenges/
               April2006.html">"Ponder This" puzzle for April, 2006</a>
%H A003149 T. Mansour and J. West, <a href="http://arXiv.org/abs/math.CO/0207204">
               Avoiding 2-letter signed patterns</a>.
%H A003149 Leroy Quet, <a href="http://www.prism-of-spirals.net/">Home Page</a> 
               (listed in lieu of email address)
%H A003149 V. Strehl, <a href="a003149.pdf">The average number of splitters in a 
               random permutation</a> [Unpublished; included here with the author's 
               permission.]
%F A003149 a(n) = n!+((n+1)/2)a(n-1), n >= 1. - Leroy Quet, Sep 06 2002
%F A003149 a(n) = ((3n+1)/2)a(n-1)-(m^2/2)a(m-2), n >= 2. - David W. Wilson, Sep 
               06, 2002
%F A003149 G.f.: (Sum_{k>=0} k!*x^k)^2. - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Aug 30 2002
%F A003149 E.g.f: log(1-x)/(x/2-1) if offset 1.
%F A003149 Convolution of A000142 [factorial numbers] with itself - Ross La Haye 
               (rlahaye(AT)new.rr.com), Oct 29 2004
%F A003149 a(n)=Sum(k*A145878(n+1,k),k=0..n+1). [From Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Oct 28 2008]
%F A003149 a(n)=A084938(n+2,2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 17 2008]
%o A003149 (PARI) a(n)=sum(k=0,n,k!*(n-k)!)
%o A003149 (PARI) a(n)=if(n<0,0,(n+1)!*polcoeff(log(1-x+x^2*O(x^n))/(x/2-1),n+1))
%Y A003149 Cf. A046825, A046878, A046879.
%Y A003149 A052186, A006932, A145878 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Oct 28 2008]
%Y A003149 Sequence in context: A000112 A127083 A131178 this_sequence A027046 A000522 
               A007469
%Y A003149 Adjacent sequences: A003146 A003147 A003148 this_sequence A003150 A003151 
               A003152
%K A003149 nonn,easy,nice
%O A003149 0,2
%A A003149 N. J. A. Sloane (njas(AT)research.att.com), H. W. Gould
%E A003149 More terms from Michel ten Voorde (seqfan(AT)tenvoorde.org) Apr 11 2001
%E A003149 Additional comments from Michael Somos, Feb 14, 2002

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