OFFSET
3,2
COMMENTS
a(n) is the number of ways in which a deck with n - 1 matched pairs and two singleton cards may be dealt into n hands of two cards, assuming the order of the hands and the order of the cards in each hand is irrelevant. (See Art of Problem Solving link for proof.) - Joel B. Lewis, Sep 30 2012
REFERENCES
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).
LINKS
John Cerkan, Table of n, a(n) for n = 3..450
Art of Problem Solving, Partitioning a deck with 2 cards in n types into pairs
P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41.
FORMULA
a(n) ~ 2^(3/2) * n^(n-2) / exp(n-3/4). - Vaclav Kotesovec, Apr 27 2015
EXAMPLE
For n = 3, the a(3) = 6 ways to partition the deck {1, 1, 2, 2, 3, 4} into three pairs are {11, 22, 34}, {12, 12, 34}, {13, 14, 22}, {11, 23, 24}, {12, 13, 24} and {12, 14, 23}. - Joel B. Lewis, Sep 30 2012
PROG
(PARI)
/* b(n) := A002135(n) */
b(n) = if(n<3, [1, 1, 2][n+1], n*b(n-1) - (n-1)*(n-2)*b(n-3)/2 );
c(n) = if(n<3, [1, 2][n], b(n-1) + (n-1)*b(n-2) + (n-1)*(n-2)*c(n-2) );
a(n) = c(n-2);
/* Joerg Arndt, Apr 07 2013 */
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Added more terms, Joerg Arndt, Apr 07 2013
STATUS
approved