A371698 - OEIS

%I #18 Apr 25 2024 13:52:37

%S 2,9,54,323,1848,10293,56738,312327,1723692,9549785,53121654,

%T 296593547,1661423104,9333552509,52565738570,296696569871,

%U 1677887732820,9505147063713,53928737011358,306393222740883,1742919983985192,9925790283119429,56584658970159474,322879453747840023

%N Number of partial order-preserving or -reversing transformations of a chain of length n.

%H V. H. Fernandes, G. M. S. Gomes, and M. M. Jesus, <a href="http://dx.doi.org/10.1081/AGB-200047446">Presentations for some monoids of partial transformations on a finite chain</a>, Communications in Algebra, 33 (2005), 587-604, 2005.

%F a(n) = 4*Sum_{k=0..n-1} binomial(n-1, k)*binomial(n+k, k) - (1 + n*(2 ^ n - 1)).

%F a(n) = 2*A002003(n) - (1 + n*(2^n - 1)).

%o (GAP) n -> 4 * Sum([0 .. n - 1], k ->  Binomial(n - 1, k) * Binomial(n + k, k)) - (1 + n * (2 ^ n - 1));

%o (PARI) a(n) = 4 * sum(k=0, n-1, binomial(n -1, k)*binomial(n + k, k)) - (1 + n * (2 ^ n - 1)); \\ _Michel Marcus_, Apr 03 2024

%Y Cf. A002003.

%K nonn

%O 1,1

%A _James Mitchell_, Apr 03 2024