%I #28 May 01 2023 10:00:53

%S 1,0,6,72,910,12000,163086,2266544,32043726,459167040,6651400756,

%T 97214919648,1431514320886,21213380196736,316072831033350,

%U 4731683468079072,71128104013487310,1073134004384407680,16243463355081280080,246585461357885877920

%N a(n) = Sum_{k = 0..n} (-1)^(n + k)*binomial(n + k - 1, k)^2.

%C It is known that Sum_{k = 0..2*n} (-1)^(n+k)*C(2*n,k)^2 = C(2*n,n). Here, we consider by analogy Sum_{k = 0..n} (-1)^(n+k)*C(-n,k)^2, where C(-n,k) = (-1)^k * C(n+k-1,k) for integer n and nonnegative integer k.

%C The sequence b(n) = C(2*n,n) of central binomial coefficients satisfies the supercongruences b(n*p^k) = b(n*p^(k-1)) ( mod p^(3*k) ) for all prime p >= 5 and any positive integers n and k - see Mestrovic. We conjecture that the present sequence also satisfies these congruences. Some examples of the congruences are given below. For a proof of the particular case of these congruences, a(p) == 0 ( mod p^3 ) for prime p >= 5, see the Bala link. The sequence a(p)/(2*p^3) for prime p >= 5 begins [48, 304, 36519504, 4827806144, 109213719151680, 17975321574419440, ...]. Cf. A034602.

%C More generally, calculation suggests that for positive integer A and integer B, the sequence a(A,B;n) := Sum_{k = 0..A*n} (-1)^(n+k)* C(-B*n,k)^2 = Sum_{k = 0..A*n} (-1)^(n+k)*C(B*n+k-1,k)^2 may also satisfy the above congruences for all prime p >= 5.

%H Seiichi Manyama, <a href="/A333593/b333593.txt">Table of n, a(n) for n = 0..500</a>

%H Peter Bala, <a href="/A333593/a333593.pdf">Notes on A333593</a>

%H R. Mestrovic, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv:1111.3057 [math.NT], 2011.

%F a(n) ~ 2^(4*n) / (5*Pi*n). - _Vaclav Kotesovec_, Mar 28 2020

%e Examples of congruences:

%e a(11) = 97214919648 = (2^5)*3*(7^2)*(11^3)*15527 == 0 ( mod 11^3 ).

%e a(2*7) - a(2) = 316072831033350 - 6 = (2^13)*3*(7^3)*11*691*4933 == 0 ( mod 7^3 ).

%e a(5^2) - a(5) = 3164395891098711251676512000 - 12000 = (2^5)*(5^6)*29* 124891891*1747384859327 == 0 ( mod 5^6 ).

%p seq( add( (-1)^(n+k)*binomial(n+k-1,k)^2, k = 0..n ), n = 0..25);

%t Table[Binomial[2*n-1, n]^2 * HypergeometricPFQ[{1, -n, -n}, {1 - 2 n, 1 - 2 n}, -1], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 28 2020 *)

%o (PARI) a(n) = sum(k=0, n, (-1)^(n+k)*binomial(n+k-1, k)^2); \\ _Michel Marcus_, Mar 29 2020

%Y Cf. A000984, A034602, A333592.

%K nonn,easy

%O 0,3

%A _Peter Bala_, Mar 27 2020