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A000459
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Number of permutations with no hits on 2 main diagonals.
(Formerly M4750 N2032)
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4
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0, 1, 10, 297, 13756, 925705, 85394646, 10351036465, 1596005408152, 305104214112561, 70830194649795010, 19629681235869138841, 6401745422388206166420, 2427004973632598297444857, 1058435896607583305978409166
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Card-matching numbers (Dinner-Diner matching numbers): A deck has n kinds of cards, 2 of each kind. The deck is shuffled and dealt in to n hands with 2 cards each. A match occurs for every card in the j-th hand of kind j. A(n) is the number of ways of achieving no matches. The probability of no matches is A(n)/((2n)!/2!^n).
Number of permutations of (0 to infinity) distinct letters (ABC......XYW etc.) each with 2 copies such that free (0) remain fixed points. E.g. if AABBCCDDEEFF (2*6=12 letters) then free (0) fixed points n5=925705 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 02 2006
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REFERENCES
| F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching Problem, Mathematics Magazine 69 (1996), 190-197.
B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), 107-118.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 187.
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
| T. D. Noe, Table of n, a(n) for n=1..100
Index entries for sequences related to card matching
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FORMULA
| G.f.: sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k) where n is the number of kinds of cards, k is the number of cards of each kind (2 in this case) and R(x, n, k) is the rook polynomial given by R(x, n, k)=(k!^2*sum(x^j/((k-j)!^2*j!))^n (see Riordan). coeff(R(x, n, k), x, j) indicates the coefficient for x^j of the rook polynomial.
a[n]=n(2n-1)*a[n-1]+2n(n-1)a[n-2]-(2n-1), a[1]=0, a[2]=1
a(n) = round(2^(n/2 + 3/4)*Pi^(-1/2)*exp(-2)*n!*BesselK(1/2+n,2^(1/2))) - Mark van Hoeij, Oct 30 2011.
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EXAMPLE
| There are 297 ways of achieving zero matches when there are 2 cards of each kind and 4 kinds of card so A(4)=297.
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MAPLE
| p := (x, k)->k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)->p(x, k)^n; f := (t, n, k)->sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k); seq(f(0, n, 2)/2!^n, n=0..18);
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CROSSREFS
| Equals A000316/2^n.
Cf. A008290, A059056-A059071, A033581.
See A059072 for another version.
Sequence in context: A024295 A159960 A059072 * A125288 A173479 A173478
Adjacent sequences: A000456 A000457 A000458 * A000460 A000461 A000462
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Formulae, more terms, etc. from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 22 2000
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