A002136 Matrices with 2 rows.

1, 2, 6, 23, 109, 618, 4096, 31133, 267219, 2557502, 27011734, 312115953, 3916844779, 53053052462, 771450742596, 11986779006647, 198204672604489, 3475110017769282, 64396888392712366, 1257612452945760503, 25815617698822423341, 555708180579477963962, 12517189538209383465496

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn

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

Examples

			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
		

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.

Programs

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 */

Formula

a(n+1) = A002135(n) + n*A002135(n - 1) + n*(n - 1)*a(n - 1). - Joel B. Lewis, Sep 30 2012

a(n) ~ 2^(3/2) * n^(n-2) / exp(n-3/4). - Vaclav Kotesovec, Apr 27 2015

Extensions

Added more terms, Joerg Arndt, Apr 07 2013