OFFSET
1,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..830
Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, Enumeration of parallelogram polycubes, arXiv:2105.00971 [cs.DM], 2021.
FORMULA
a(n) = Sum_{m=1..n} (binomial(n - 1, m - 1)*binomial(n, m - 1)/m)^2.
a(n) = 4F3([1 - n, 1 - n, - n, - n], [1, 2, 2], 1), where F is the generalized hypergeometric function.
From Vaclav Kotesovec, Dec 24 2018: (Start)
Recurrence: n*(n+1)^3*(5*n^2 - 10*n + 4)*a(n) = 2*n*(2*n - 1)*(15*n^4 - 30*n^3 + 7*n^2 + 8*n - 8)*a(n-1) + 4*(n-2)^2*(4*n - 5)*(4*n - 3)*(5*n^2 - 1)*a(n-2).
a(n) ~ 2^(4*n + 1/2) / (Pi^(3/2) * n^(7/2)).
(End)
MAPLE
a := n -> add(binomial(n-1, m-1)^2*binomial(n, m-1)^2/m^2, m = 1 .. n): seq(a(n), n = 1 .. 20)
MATHEMATICA
Table[HypergeometricPFQ[{1-n, 1-n, -n, -n}, {1, 2, 2}, 1], {n, 1, 20}]
PROG
(GAP) List([1..20], n->Sum([1..n], m->(Binomial(n-1, m-1)*Binomial(n, m-1)/m)^2));
(PARI) a(n) = sum(m=1, n, (binomial(n-1, m-1)*binomial(n, m-1)/m)^2);
(Sage) [hypergeometric([-n, -n, -n+1, -n+1], [1, 2, 2], 1).simplify_hypergeometric() for n in (1..25)] # G. C. Greubel, Feb 15 2021
(Magma) [(&+[(Binomial(n-1, j-1)*Binomial(n, j-1)/j)^2 : j in [1..n]]): n in [1..25]]; // G. C. Greubel, Feb 15 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Stefano Spezia, Dec 23 2018
STATUS
approved