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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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7
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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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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
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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, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Gejza Jenca and Peter Sarkoci, Linear extensions and order-preserving poset partitions, arXiv:1112.5782 [math.CO], 2011-2015. - 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_{i, j = 1...(n-1), i+j = n} binomial(n, i)*binomial(n, j).
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, A115111, A115246.
Sequence in context: A022728 A231950 A246134 * A171074 A005367 A050184
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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