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A115112
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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.
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4
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0, 4, 18, 68, 250, 922, 3430, 12868, 48618, 184754, 705430, 2704154, 10400598, 40116598, 155117518, 601080388, 2333606218, 9075135298, 35345263798, 137846528818, 538257874438, 2104098963718, 8233430727598, 32247603683098
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OFFSET
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1,2
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COMMENTS
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The 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.
The 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
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..300
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
Gejza Jenca and Peter Sarkoci, Linear extensions and order-preserving poset partitions, arXiv preprint arXiv:1112.5782, 2011. - From N. J. A. Sloane, Apr 08 2012
Ran Pan, Exercise K, Project P.
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FORMULA
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a(n) = binomial(2*n, n)-2 = A000984(n)-2; also: a(n)=sum{binomial(n, i)*binomial(n, j|i, j=1...(n-1), i+j=n}.
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
a(n) ~ 4^n/sqrt(Pi*n). - Vaclav Kotesovec, Oct 19 2012
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EXAMPLE
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a(5)=binomial(10,5)-2=250.
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MAPLE
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seq(sum((binomial(n, m))^2, m=1..n-1), n=1..24); # Zerinvary Lajos, Jun 19 2008
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MATHEMATICA
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Table[Sum[Binomial[n, i] Binomial[n, n - i], {i, 1, n - 1}], {n, 1, 10}]
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PROG
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(MAGMA) [Binomial(2*n, n)-2: n in [1..25]]; // Vincenzo Librandi, Apr 10 2015
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CROSSREFS
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Cf. A000984, A115246, A115111.
Sequence in context: A231950 A246134 * A171074 A005367 A050184 A263582
Adjacent sequences: A115109 A115110 A115111 * A115113 A115114 A115115
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KEYWORD
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nonn,easy
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AUTHOR
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Hieronymus Fischer, Jan 22 2006
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STATUS
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approved
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