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A002136 Matrices with 2 rows.
(Formerly M1668 N0656)
1
1, 2, 6, 23, 109, 618, 4096, 31133, 267219, 2557502, 27011734, 312115953, 3916844779, 53053052462, 771450742596, 11986779006647, 198204672604489, 3475110017769282, 64396888392712366, 1257612452945760503, 25815617698822423341, 555708180579477963962, 12517189538209383465496 (list; graph; refs; listen; history; text; internal format)
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+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

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

Sequence in context: A112501 A093345 A289681 * A208733 A264899 A224786

Adjacent sequences:  A002133 A002134 A002135 * A002137 A002138 A002139

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Added more terms, Joerg Arndt, Apr 07 2013

STATUS

approved

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Last modified January 23 01:45 EST 2019. Contains 319365 sequences. (Running on oeis4.)