A366449 Number of smooth discrete aggregation functions defined on the finite chain L_n={0,1,...,n-1,n} having neutral element/absorbing element.

2, 5, 18, 102, 970, 15947, 453872, 22174642, 1846384884, 260939482721, 62454382216334, 25285347265901814, 17304115945924822724, 20008412370393070905186, 39078178288867371807316956, 128893469663525965017925474046, 717867336460661639426421067202992, 6750439274904330523572066561554305664

Offset: 1

Author

Marc Munar, Oct 10 2023

Keywords

• nonn
• easy

Comments

The number of smooth discrete aggregation functions on the finite chain L_n={0,1,...,n-1,n} having neutral element/absorbing element e\in L_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 (discrete aggregation function); F(x+1,y)-F(x,y)<=1 and F(x,y+1)-F(x,y)<= 1 (smooth); and F(x,a)=F(a,x)=x (neutral element) or F(x,a)=F(a,x)=a (absorbing element).

Crossrefs

Cf. A005130.

Programs

Mathematica

Table[2*Product[Factorial[3 i + 1]/Factorial[n + i], {i, 0, n - 1}] +
  Sum[Product[Factorial[3 i + 1]/Factorial[a + i], {i, 0, a - 1}]*
    Product[Factorial[3 i + 1]/Factorial[n - a + i], {i, 0, n - a - 1}], {a, 1, n - 1}], {n, 1, 13}]

Formula

a(n) = 2*Product_{i=0..n-1} (3i+1)!/(n+i)!+ Sum_{a=1..n-1}(Product_{i=0..a-1} (3i+1)!/(a+i)!)*(Product_{i=0..n-a-1} (3i+1)!/(n-a+i)!).

a(n) ~ exp(1/36) * Pi^(1/3) * 3^(3*n^2/2 - 7/36) / (A^(1/3) * Gamma(1/3)^(2/3) * n^(5/36) * 2^(2*n^2 - 17/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 18 2023