A366446 - OEIS

%I #9 Nov 18 2023 15:18:48

%S 4,136,21238,15374304,52326659814,845020424183364,

%T 65102985676317726176,24005912569370916926920192,

%U 42445533661127789112292364580400,360256756545210313397342412375573121875,14686785417300337272290307023148088973414062500

%N Number of discrete aggregation functions defined on the finite chain L_n={0,1,...,n-1,n}.

%C The number of discrete aggregation functions on the finite chain L_n={0,1,...,n-1,n}, i.e., the number of monotonic increasing binary functions F: L_n^2->L_n such that F(0,0)=0 and F(n,n)=n.

%H M. Munar, S. Massanet and D. Ruiz-Aguilera, <a href="https://doi.org/10.1016/j.ins.2022.10.121">On the cardinality of some families of discrete connectives</a>, Information Sciences, Volume 621, 2023, 708-728.

%F a(n) = Product_{i=1..n+1} Product_{j=1..n+1} Product_{k=1..n} (i+j+k-1)/(i+j+k-2) - 2*Product_{i=1..n+1} Product_{j=1..n+1} Product_{k=1..n-1} (i+j+k-1)/(i+j+k-2) + Product_{i=1..n+1} Product_{j=1..n+1} Product_{k=1..n-2} (i+j+k-1)/(i+j+k-2).

%F From _Vaclav Kotesovec_, Nov 18 2023: (Start)

%F a(n) = BarnesG(n+1)^3 * BarnesG(3*n+1) * (3*Gamma(n) * Gamma(3*n) / Gamma(2*n)^2 - 2) * (3*(3*n+1) * Gamma(n) * Gamma(3*n) / Gamma(2*n)^2 - 2*(n-1)) / (16*(2*n+1) * BarnesG(2*n+1)^3)).

%F a(n) ~ exp(1/12) * 3^(9*n^2/2 + 6*n + 23/12) / (A * n^(1/12) * 2^(6*n^2 + 8*n + 11/4)), where A is the Glaisher-Kinkelin constant A074962. (End)

%t Table[Product[

%t    Product[Product[(i + j + k - 1)/(i + j + k - 2), {k, 1, n}], {j, 1,

%t       n + 1}], {i, 1, n + 1}] -

%t   2*Product[

%t     Product[Product[(i + j + k - 1)/(

%t       i + j + k - 2), {k, 1, n - 1}], {j, 1, n + 1}], {i, 1, n + 1}] +

%t    Product[Product[

%t     Product[(i + j + k - 1)/(i + j + k - 2), {k, 1, n - 2}], {j, 1,

%t      n + 1}], {i, 1, n + 1}], {n, 2, 13}]

%t Table[BarnesG[n + 1]^3 * BarnesG[3*n + 1]*(3*Gamma[n] * Gamma[3*n]/Gamma[2*n]^2 - 2) * (3*(3*n + 1)*Gamma[n]*Gamma[3*n]/Gamma[2*n]^2 - 2*(n-1)) / (16*(2*n + 1) * BarnesG[2*n + 1]^3), {n, 1, 13}] (* _Vaclav Kotesovec_, Nov 18 2023 *)

%K nonn

%O 1,1

%A _Marc Munar_, Oct 10 2023