A214441 -id:A214441 - cp's OEIS frontend

A201555 a(n) = C(2*n^2,n^2) = A000984(n^2), where A000984 is the central binomial coefficients.

1, 2, 70, 48620, 601080390, 126410606437752, 442512540276836779204, 25477612258980856902730428600, 23951146041928082866135587776380551750, 365907784099042279561985786395502921046971688680, 90548514656103281165404177077484163874504589675413336841320

Offset: 0

Paul D. Hanna, Dec 02 2011

nonn

Central coefficients of triangle A228832.

			L.g.f.: L(x) = 2*x + 70*x^2/2 + 48620*x^3/3 + 601080390*x^4/4 + ...
where exponentiation equals the g.f. of A201556:
exp(L(x)) = 1 + 2*x + 37*x^2 + 16278*x^3 + 150303194*x^4 + ... + A201556(n)*x^n + ...

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

Cf. A000984, A007814, A159918, A201556, A214441, A228832, A285388, A285717.

Table[Binomial[2n^2,n^2],{n,0,10}] (* Harvey P. Dale, Dec 10 2011 *)

PARI
```
a(n) = binomial(2*n^2,n^2)
```

from math import comb
def A201555(n): return comb((m:=n**2)<<1,m) # Chai Wah Wu, Jul 08 2022

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.

A254746 a(n) = Catalan(n^2) mod prime(n).

Original entry on oeis.org

1, 2, 2, 5, 8, 0, 0, 11, 0, 24, 0, 0, 29, 0, 0, 0, 0, 39, 58, 0, 21, 33, 9, 69, 9, 0, 16, 86, 0, 0, 0, 0, 0, 64, 139, 0, 0, 0, 75, 12, 4, 0, 0, 119, 195, 0, 193, 202, 0, 0, 55, 218, 0, 0, 0, 0, 84, 201, 0, 0, 203, 275, 0, 198, 159, 0, 0, 0, 0, 255, 13, 204, 0

Offset: 1

Vincenzo Librandi, Feb 07 2015

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000040, A214441, A246714, A254706.

[Catalan(n^2) mod NthPrime(n): n in [1..100]];

Table[Mod[CatalanNumber[n^2], Prime[n]], {n, 80}]

from sympy import factorint, prime
A254746_list, c, s, s2 = [1], {}, 2, 4
for n in range(2,10**3+1):
    for p,e in factorint(4*n-2).items():
        if p in c:
            c[p] += e
        else:
            c[p] = e
    for p,e in factorint(n+1).items():
        if c[p] == e:
            del c[p]
        else:
            c[p] -= e
    if n == s2:
        d, ps = 1, prime(s)
        for p,e in c.items():
            d = (d*pow(p,e,ps)) % ps
        A254746_list.append(d)
        s2 += 2*s+1
        s += 1 # Chai Wah Wu, Feb 14 2015

a(n) = A000108(n^2) mod A000040(n).

cp's OEIS Frontend

A201555 a(n) = C(2*n^2,n^2) = A000984(n^2), where A000984 is the central binomial coefficients.

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