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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. 4
0, 4, 18, 68, 250, 922, 3430, 12868, 48618, 184754, 705430, 2704154, 10400598, 40116598, 155117518, 601080388, 2333606218, 9075135298, 35345263798, 137846528818, 538257874438, 2104098963718, 8233430727598, 32247603683098 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

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.

FORMULA

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

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, A115246, A115111.

Sequence in context: A231950 A246134 * A171074 A005367 A050184 A263582

Adjacent sequences:  A115109 A115110 A115111 * A115113 A115114 A115115

KEYWORD

nonn,easy

AUTHOR

Hieronymus Fischer, Jan 22 2006

STATUS

approved

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Last modified May 26 05:25 EDT 2017. Contains 287077 sequences.