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

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

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.

Formula

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

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

Cf. A000984, A115111, A115246.

Sequence in context: A022728 A231950 A246134 * A171074 A376072 A005367

Adjacent sequences: A115109 A115110 A115111 * A115113 A115114 A115115

Keyword

nonn,easy

Author

Hieronymus Fischer, Jan 22 2006

Status

approved