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a(n) = Sum_{k = 0..n} binomial(n, k)^2*binomial(n+k, k)*binomial(3*n+2*k+1, n).
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%I #11 Jul 22 2024 15:21:44

%S 1,16,783,52000,3984625,331782528,29167556544,2664193232448,

%T 250366507021125,24050203166498080,2350951602405723031,

%U 233101029618601148160,23386851829632443099584,2369867535548685346720000,242199626637223381868264640,24935593829720710016846519424,2583792728153680286245574534061

%N a(n) = Sum_{k = 0..n} binomial(n, k)^2*binomial(n+k, k)*binomial(3*n+2*k+1, n).

%C A companion sequence to A374605.

%C Conjecture: for prime p, a(n) is divisible by p^3 for integer n in the interval [(1/3)*(2*p + (p/3)), p - 1], where (p/3) denotes the Legendre symbol.

%C More generally, for m >= 2, a similar divisibility property appears to hold for the sequence whose n-th term is equal to Sum_{k = 0..n} binomial(n, k)^2* binomial(n+k, k)*binomial((m + 1)*n + m*k + 1, n).

%F a(n) = binomial(3*n+1,n) * hypergeom([-n, -n, (3*n+2)/2, (3*n+3)/2], [1, 1, n+3/2], 1).

%F P-recursive: 16*(5616*n^4 - 22464*n^3 + 33183*n^2 - 21438*n + 5116)*(4*n - 1)^2*(4*n + 1)^2*n^3*a(n) = 36*(36391680*n^10 - 181958400*n^9 + 367112736*n^8 - 376700544*n^7 + 196483887*n^6 - 35225037*n^5 - 11205753*n^4 + 5552733*n^3 - 287502*n^2 - 163800*n + 20800)*(2*n - 1)*a(n-1) + 27*(n - 1)*(5616*n^4 - 513*n^2 + 13)*(3*n - 2)^3*(3*n - 4)^3*a(n-2) with a(0) = 1 and a(1) = 16.

%F a(n) ~ 3^((9*n + 3)/2) * (1 + sqrt(3))^(6*n+3) / (Pi^(3/2) * n^(3/2) * 2^(9*n + 13/2)). - _Vaclav Kotesovec_, Jul 16 2024

%e Factorization of a(9) thru a(12) showing divisibility by 13^3:

%e a(9) = (2^5)*5*(11^3)*(13^3)*1109*46351

%e a(10) = (11^3)*(13^3)*803962100633

%e a(11) = (2^8)*(3^3)*5*(13^3)*(17^3)*67*79*118057

%e a(12) = (2^6)*7*(13^3)*(17^3)*4836341163053.

%p seq( add(binomial(n, k)^2*binomial(n+k, k)*binomial(3*n+2*k+1, n), k = 0..n), n = 0..20);

%p # faster program for large n

%p seq(simplify( binomial(3*n+1,n) * hypergeom([-n, -n, (3*n+2)/2, (3*n+3)/2], [1, 1, n+3/2], 1)), n = 0..20);

%Y Cf. A176285, A374605.

%K nonn,easy

%O 0,2

%A _Peter Bala_, Jul 13 2024