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

1,1

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).

Links

Table of n, a(n) for n=1..18.

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

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}]

Crossrefs

Cf. A005130.

Sequence in context: A279207 A005639 A093730 * A304918 A007769 A174122

Adjacent sequences: A366446 A366447 A366448 * A366450 A366451 A366452

Keyword

nonn,easy

Author

Marc Munar, Oct 10 2023

Status

approved