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.
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
Examples
a(5) = binomial(10,5) - 2 = 250.
Links
- 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.
Programs
-
Magma
[Binomial(2*n, n)-2: n in [1..25]]; // Vincenzo Librandi, Apr 10 2015
-
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}]
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
Comments