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A002136 Matrices with 2 rows.

%I A002136 M1668 N0656 #36 May 01 2025 01:34:57
%S A002136 1,2,6,23,109,618,4096,31133,267219,2557502,27011734,312115953,
%T A002136 3916844779,53053052462,771450742596,11986779006647,198204672604489,
%U A002136 3475110017769282,64396888392712366,1257612452945760503,25815617698822423341,555708180579477963962,12517189538209383465496
%N A002136 Matrices with 2 rows.
%C A002136 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
%D A002136 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002136 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002136 John Cerkan, <a href="/A002136/b002136.txt">Table of n, a(n) for n = 3..450</a>
%H A002136 Art of Problem Solving, <a href="https://artofproblemsolving.com/community/c6h366633">Partitioning a deck with 2 cards in n types into pairs</a>
%H A002136 P. A. MacMahon, <a href="http://plms.oxfordjournals.org/content/s2-17/1/25.extract">Combinations derived from m identical sets of n different letters and their connexion with general magic squares</a>, Proc. London Math. Soc., 17 (1917), 25-41.
%F A002136 a(n+1) = A002135(n) + n*A002135(n - 1) + n*(n - 1)*a(n - 1). - _Joel B. Lewis_, Sep 30 2012
%F A002136 a(n) ~ 2^(3/2) * n^(n-2) / exp(n-3/4). - _Vaclav Kotesovec_, Apr 27 2015
%e A002136 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
%o A002136 (PARI)
%o A002136 /* b(n) := A002135(n) */
%o A002136 b(n) = if(n<3, [1,1,2][n+1], n*b(n-1) - (n-1)*(n-2)*b(n-3)/2 );
%o A002136 c(n) = if(n<3, [1,2][n], b(n-1) + (n-1)*b(n-2) + (n-1)*(n-2)*c(n-2) );
%o A002136 a(n) = c(n-2);
%o A002136 /* _Joerg Arndt_, Apr 07 2013 */
%K A002136 nonn
%O A002136 3,2
%A A002136 _N. J. A. Sloane_
%E A002136 Added more terms, _Joerg Arndt_, Apr 07 2013