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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 (list; graph; refs; listen; history; text; internal format)
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 A005367 A050184

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 December 7 13:08 EST 2021. Contains 349581 sequences. (Running on oeis4.)