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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 Inscribed-Triangle 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! * (n-i)!)^2.

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

MATHEMATICA

Table[Sum[(k!*(n-k)!)^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

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Last modified October 16 07:07 EDT 2021. Contains 348041 sequences. (Running on oeis4.)