OFFSET
0,2
COMMENTS
Central coefficients of triangle A228832.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..40
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
R. Oblath, Congruences with binomial coefficients, Proceedings of the Indian Academy of Science, Section A, Vol. 1 No. 6, 383-386
FORMULA
L.g.f.: ignoring initial term, equals the logarithmic derivative of A201556.
a(n) = (2*n^2)! / (n^2)!^2.
a(n) = Sum_{k=0..n^2} binomial(n^2,k)^2.
For primes p >= 5: a(p) == 2 (mod p^3), Oblath, Corollary II; a(p) == binomial(2*p,p) (mod p^6) - see Mestrovic, Section 5, equation 31. - Peter Bala, Dec 28 2014
A007814(a(n)) = A159918(n). - Antti Karttunen, Apr 27 2017, based on Vladimir Shevelev's Jul 20 2009 formula in A000984.
EXAMPLE
MATHEMATICA
Table[Binomial[2n^2, n^2], {n, 0, 10}] (* Harvey P. Dale, Dec 10 2011 *)
PROG
(PARI) a(n) = binomial(2*n^2, n^2)
(Python)
from math import comb
def A201555(n): return comb((m:=n**2)<<1, m) # Chai Wah Wu, Jul 08 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 02 2011
STATUS
approved