A272913 a(n) = 3*2^(2*n-1)*(n-1)*n^3*binomial(2*n,n)^2, a closed form for a triple binomial sum involving absolute values.

0, 0, 6912, 2073600, 361267200, 48771072000, 5665247723520, 595732271726592, 58357447026278400, 5420989989833932800, 483204292920999936000, 41671538221507034480640, 3497929581885972295974912, 287077554068924493987840000, 23115688495680026711162880000

Offset

0,3

Comments

See Theorem 6 of Brent et al. article.

a(n) is divisible by 48^2.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, Helmut Prodinger, Some binomial sums involving absolute values, arXiv:1411.1477 [math.CO], 2016, page 11.

Formula

a(n) = Sum_{i=-n..n} (Sum_{j=-n..n} (Sum_{k=-n..n} binomial(2*n, n+i)*binomial(2*n, n+j)*binomial(2*n, n+k)*|(i^2-j^2)*(i^2-k^2)*(j^2-k^2)|)).

G.f.: 6912*x^2*(2F1(5/2, 5/2, 2, 64*x) + 100*x*2F1(7/2, 7/2, 3, 64*x)), where 2F1() is the Gauss hypergeometric function.

D-finite with recurrence (n-2)*(n-1)^2*a(n) -16*n*(2*n-1)^2*a(n-1)=0. - R. J. Mathar, Feb 08 2021

Mathematica

Table[3 2^(2 n - 1) (n - 1) n^3 Binomial[2 n, n]^2, {n, 0, 20}]

Prog

(PARI) vector(20, n, n--; 3*2^(2*n-1)*(n-1)*n^3*binomial(2*n, n)^2)
(Sage) [3*2^(2*n-1)*(n-1)*n^3*binomial(2*n, n)^2 for n in range(20)]
(Magma) [3*2^(2*n-1)*(n-1)*n^3*Binomial(2*n, n)^2: n in [0..20]];

Crossrefs

Cf. A254408, A268147, A268148, A268149, A268150, A268151, A268152, A269877.

Sequence in context: A156419 A268908 A192077 * A230468 A186059 A253612

Adjacent sequences: A272910 A272911 A272912 * A272914 A272915 A272916

Keyword

nonn,easy

Author

Bruno Berselli, May 10 2016

Status

approved