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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.
7
0, 4, 18, 68, 250, 922, 3430, 12868, 48618, 184754, 705430, 2704154, 10400598, 40116598, 155117518, 601080388, 2333606218, 9075135298, 35345263798, 137846528818, 538257874438, 2104098963718, 8233430727598, 32247603683098, 126410606437750, 495918532948102
OFFSET
1,2
COMMENTS
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
LINKS
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.
FORMULA
a(n) = binomial(2*n, n) - 2 = A000984(n) - 2.
a(n) = Sum_{i=1..n-1} binomial(n,i)^2.
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
E.g.f.: exp(2*x) * BesselI(0,2*x) - 2*exp(x) + 1. - Ilya Gutkovskiy, Mar 04 2021
EXAMPLE
a(5) = binomial(10,5) - 2 = 250.
MAPLE
seq(sum((binomial(n, m))^2, m=1..n-1), n=1..24); # Zerinvary Lajos, Jun 19 2008
MATHEMATICA
Table[Sum[Binomial[n, i] Binomial[n, n - i], {i, 1, n - 1}], {n, 1, 10}]
PROG
(Magma) [Binomial(2*n, n)-2: n in [1..25]]; // Vincenzo Librandi, Apr 10 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Hieronymus Fischer, Jan 22 2006
STATUS
approved