A279055 Self-convolution of squares of factorial numbers (A001044).

1, 2, 9, 80, 1240, 30240, 1071504, 51996672, 3307723776, 266872320000, 26615381760000, 3214252921651200, 462189467175321600, 78024380924038348800, 15279632043682406400000, 3435553774431004262400000, 879010223384483132866560000, 253916900613208108255150080000

Offset: 0

Author

Arman Maesumi, Dec 04 2016

Keywords

• nonn

Comments

a(n) = (n!)^2 * Sum_{i=0..n} (binomial(n,i)^(-2)).

Consider a triangle ABC with area p. Let points X, Y, Z be randomly and uniformly chosen on sides BC, CA, BA. Let r = area of XYZ. Then the average or expected value of (r/p)^n = a(n)/(n!^2 * (n+1)^3).

a(n) = (3*(n+1)^4 *(n!)^4 /(2n+3)!) * Sum_{i=1..n+1} ((1/i)* binomial(2i, i)), see Sprugnoli Formula 5.2 as noted by Markus Scheuer.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..253
• Arman Maesumi, Triangle Inscribed-Triangle Picking, arXiv:1804.11007 [math.GM], 2018.
• R. Sprugnoli, Riordan Array Proofs of Identities in Gould's Book.

Crossrefs

Cf. A000142, A001044, A003149, A100516, A100517.

Programs

Mathematica

Table[Sum[(k!*(n-k)!)^2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 05 2016 *)

Formula

a(n) = Sum_{i=0..n} (i! * (n-i)!)^2.

a(n) ~ 2*(n!)^2. - Vaclav Kotesovec, Dec 05 2016

a(n) = A001044(n)*A100516(n)/A100517(n). - Alois P. Heinz, Feb 21 2023

Extensions

Definition clarified by Georg Fischer, Feb 21 2023