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A115112 Number of different ways to select n elements from two sets of n elements under the precondition of choosing at least one element from each set.

%I A115112 #80 Feb 18 2025 10:06:01
%S A115112 0,4,18,68,250,922,3430,12868,48618,184754,705430,2704154,10400598,
%T A115112 40116598,155117518,601080388,2333606218,9075135298,35345263798,
%U A115112 137846528818,538257874438,2104098963718,8233430727598,32247603683098,126410606437750,495918532948102
%N A115112 Number of different ways to select n elements from two sets of n elements under the precondition of choosing at least one element from each set.
%C A115112 Also number of lattice paths from (0,0) to (n,n) that use steps (1,0) and (0,1) and do not include (n,0) or (0,n). - _Ran Pan_, Apr 10 2015
%H A115112 Vincenzo Librandi, <a href="/A115112/b115112.txt">Table of n, a(n) for n = 1..300</a>
%H A115112 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H A115112 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A115112 Gejza Jenca and Peter Sarkoci, <a href="http://arxiv.org/abs/1112.5782">Linear extensions and order-preserving poset partitions</a>, arXiv:1112.5782 [math.CO], 2011-2015. - From _N. J. A. Sloane_, Apr 08 2012
%H A115112 Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/warmups/eK.html">Exercise K</a>, Project P.
%F A115112 a(n) = binomial(2*n, n) - 2 = A000984(n) - 2.
%F A115112 a(n) = Sum_{i=1..n-1} binomial(n,i)^2.
%F A115112 Recurrence: n*(3*n - 5)*a(n) = (15*n^2 - 31*n + 12)*a(n-1) - 2*(2*n - 3)*(3*n - 2)*a(n-2). - _Vaclav Kotesovec_, Oct 19 2012
%F A115112 a(n) ~ 4^n/sqrt(Pi*n). - _Vaclav Kotesovec_, Oct 19 2012
%F A115112 E.g.f.: exp(2*x) * BesselI(0,2*x) - 2*exp(x) + 1. - _Ilya Gutkovskiy_, Mar 04 2021
%e A115112 a(5) = binomial(10,5) - 2 = 250.
%p A115112 seq(sum((binomial(n,m))^2,m=1..n-1),n=1..24); # _Zerinvary Lajos_, Jun 19 2008
%t A115112 Table[Sum[Binomial[n, i] Binomial[n, n - i], {i, 1, n - 1}], {n, 1, 10}]
%o A115112 (Magma) [Binomial(2*n, n)-2: n in [1..25]]; // _Vincenzo Librandi_, Apr 10 2015
%Y A115112 Cf. A000984, A115111, A115246.
%K A115112 nonn,easy
%O A115112 1,2
%A A115112 _Hieronymus Fischer_, Jan 22 2006