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