

A279055


Convolution of squares of factorial numbers (A000142).


1



1, 2, 9, 80, 1240, 30240, 1071504, 51996672, 3307723776, 266872320000, 26615381760000, 3214252921651200, 462189467175321600, 78024380924038348800, 15279632043682406400000, 3435553774431004262400000, 879010223384483132866560000
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


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 InscribedTriangle Picking, arXiv:1804.11007 [math.GM], 2018.
R. Sprugnoli, Riordan Array Proofs of Identities in Gould's Book.


FORMULA

a(n) = Sum_{i=0..n} (i! * (ni)!)^2.
a(n) ~ 2*(n!)^2.  Vaclav Kotesovec, Dec 05 2016


MATHEMATICA

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


CROSSREFS

Cf. A003149.
Sequence in context: A122720 A109519 A193208 * A320946 A135868 A212271
Adjacent sequences: A279052 A279053 A279054 * A279056 A279057 A279058


KEYWORD

nonn


AUTHOR

Arman Maesumi, Dec 04 2016


STATUS

approved



