%I A059058
%S A059058 1,0,0,0,1,1,0,9,0,9,0,1,56,216,378,435,324,189,54,27,0,1,13833,49464,
%T A059058 84510,90944,69039,38448,16476,5184,1431,216,54,0,1,6699824,23123880,
%U A059058 38358540,40563765,30573900,17399178,7723640
%N A059058 Card-matching numbers (Dinner-Diner matching numbers).
%C A059058 This is a triangle of card matching numbers. A deck has n kinds of cards, 
               3 of each kind. The deck is shuffled and dealt in to n hands with 
               3 cards each. A match occurs for every card in the j-th hand of kind 
               j. Triangle T(n,k) is the number of ways of achieving exactly k matches 
               (k=0..3n). The probability of exactly k matches is T(n,k)/((3n)!/
               (3!)^n).
%C A059058 Rows have lengths 1,4,7,10,...
%C A059058 Analogous to A008290 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 22 2005
%D A059058 F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, 
               Ch. 7 and Ch. 12.
%D A059058 F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching 
               Problem, Mathematics Magazine 69 (1996), 190-197.
%D A059058 B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 
               76 (2003), 107-118.
%D A059058 S. G. Penrice, Derangements, permanents and Christmas presents, The American 
               Mathematical Monthly 98(1991), 617-620.
%D A059058 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 
               174-178.
%D A059058 R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University 
               Press, 1997, p. 71.
%H A059058 Barbara H. Margolius, <a href="http://academic.csuohio.edu/bmargolius/
               homepage/dinner/dinner/cardentry.htm">Dinner-Diner Matching Probabilities</
               a>
%H A059058 <a href="Sindx_Ca.html#cardmatch">Index entries for sequences related 
               to card matching</a>
%F A059058 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 (here k is 3) 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 Stanley or Riordan). 
               coeff(R(x, n, k), x, j) indicates the j-th coefficient on x of the 
               rook polynomial.
%e A059058 There are 9 ways of matching exactly 2 cards when there are 2 different 
               kinds of cards, 3 of each in each of the two decks so T(2,2)=9.
%p A059058 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);
%p A059058 for n from 0 to 6 do seq(coeff(f(t,n,3),t,m)/3!^n,m=0..3*n); od;
%Y A059058 Cf. A008290, A059056-A059071.
%Y A059058 Cf. A008290.
%Y A059058 Sequence in context: A154185 A086199 A167545 this_sequence A021015 A010680 
               A131566
%Y A059058 Adjacent sequences: A059055 A059056 A059057 this_sequence A059059 A059060 
               A059061
%K A059058 nonn,tabf,nice
%O A059058 0,8
%A A059058 Barbara Haas Margolius (margolius(AT)math.csuohio.edu)