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A003149
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Sum_{k=0..n} k!(n-k)!.
(Formerly M1496)
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17
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1, 2, 5, 16, 64, 312, 1812, 12288, 95616, 840960, 8254080, 89441280, 1060369920, 13649610240, 189550368000, 2824077312000, 44927447040000, 760034451456000, 13622700994560000, 257872110354432000, 5140559166898176000
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OFFSET
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0,2
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COMMENTS
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The sequence (origin 1) is the resistance between opposite corners of an n-dimensional hypercube of unit resistors, multiplied by n!.
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 ...
Number of {12,21*,2*1}-avoiding signed permutations in the hyperoctahedral group.
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
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, Nov 02 2005
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, Apr 02 2006
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, Oct 28 2008]
It appears that a(n)=2^(-n)*(A103213(n)+n!H(n)) with H(n) harmonic number of order n.
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REFERENCES
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F. Nedemeyer and Y. Smorodinsky, Resistance in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.
Problem E3467, Amer. Math. Monthly, 100 (1993), 800-801. [From Emeric Deutsch, Oct 28 2008]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. Sprugnoli, Moments of Reciprocals of Binomial Coefficients, Journal of Integer Sequences, 14 (2011), #11.7.8.
Stanley, R. P., Enumerative Combinatorics, Volume 1 (1986), p. 49. [From Emeric Deutsch, Oct 28 2008]
B. Sury, Sum of the reciprocals of the binomial coefficients, Europ. J. Comb., 14 (1993), 351-353.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..100
Fred Curtis, Resistance-network Problems.
IBM, "Ponder This" puzzle for April, 2006
T. Mansour and J. West, Avoiding 2-letter signed patterns.
V. Strehl, The average number of splitters in a random permutation [Unpublished; included here with the author's permission.]
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FORMULA
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a(n) = n!+((n+1)/2)a(n-1), n >= 1. - Leroy Quet, Sep 06 2002
a(n) = ((3n+1)/2)a(n-1)-(n^2/2)a(n-2), n >= 2. - David W. Wilson, Sep 06, 2002
G.f.: (Sum_{k>=0} k!*x^k)^2. - Vladeta Jovovic, Aug 30 2002
E.g.f: log(1-x)/(x/2-1) if offset 1.
Convolution of A000142 [factorial numbers] with itself - Ross La Haye (rlahaye(AT)new.rr.com), Oct 29 2004
a(n) = Sum(k*A145878(n+1,k),k=0..n+1). [From Emeric Deutsch, Oct 28 2008]
a(n) = A084938(n+2,2). [From Philippe DELEHAM, Dec 17 2008]
a(n) = 2*int(Ei(t)*exp(-2*t)*t^(n+1),t=0..infinity) where Ei is the exponential integral function.
O.g.f.: 1/(1-I(x))^2 where I(x) is o.g.f. for A003319. - Geoffrey Critzer, Apr 27 2012.
a(n) ~ 2*n!. - Vaclav Kotesovec, Oct 04 2012
a(n) = (n+1)!/2^n * Sum_{k=0..n} 2^k/(k+1). - Vaclav Kotesovec, Oct 27 2012
E.g.f.: 2/((x-1)*(x-2)) + 2*x/(x-2)^2*G(0) where G(k) = 1 + x*(2*k+1)/(2*(k+1) - 4*x*(k+1)^2/(2*x*(k+1) + (2*k+3)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 14 2012
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MATHEMATICA
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Table[Sum[k!(n-k)!, {k, 0, n}], {n, 0, 20}] (* From Harvey P. Dale, Mar 28 2012 *)
Table[(n+1)!/2^n*Sum[2^k/(k+1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 27 2012 *)
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PROG
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(PARI) a(n)=sum(k=0, n, k!*(n-k)!)
(PARI) a(n)=if(n<0, 0, (n+1)!*polcoeff(log(1-x+x^2*O(x^n))/(x/2-1), n+1))
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CROSSREFS
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Cf. A046825, A046878, A046879.
Cf. A052186, A006932, A145878. - Emeric Deutsch, Oct 28 2008
Sequence in context: A185998 A127083 A131178 * A027046 A000522 A182290
Adjacent sequences: A003146 A003147 A003148 * A003150 A003151 A003152
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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, H. W. Gould
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EXTENSIONS
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More terms from Michel ten Voorde (seqfan(AT)tenvoorde.org) Apr 11 2001
Additional comments from Michael Somos, Feb 14, 2002
Second formula corrected by Naoki Sato (nsato7(AT)yahoo.ca), Jan 27 2010
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STATUS
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approved
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