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Bill Gosper, Feb 04 2004: "A few weeks ago I conjectured that
"2 binomial(n,i) (n+2i+3)! / ((i+1)!(i+2)!(n+3)!) is always an integer (summed on i, this gives the current sequence).
"This is the special case C(3,i,n-i) of C(m,k,n) :=
"(n+k)!(n+m)!/(n!(n+m+k)!) * Product_{j=1..k} (j - 1)! (n + j m + m)!/((m + j - 1)! (n + j m)!)
"which I also conjecture integral."
Alec Mihailovs, Feb 04 2004: "These conjectures are true. Consider the partition
"p(m,k,n)=(n+m,m,...,m) of n+m*(k+1), where m is repeated k times. It is easy
"to see that C(m,k,n) equals the dimension of the irreducible representation of S_(n+m*(k+1)) corresponding to p(m,k,n) calculated using hook length formula.
"Another formula for C(m,k,n) is ((n+mk+m)!/n!) * Product_{i=0..m-1} i!/((k+i)!(n+k+i+1)!)."
Bill Gosper, Mar 19, 2004: Cloitre has characterized the sequence mods 2 and 3. Remarkably, a(9k+6) mod 3 = 2*A014578(k+1), the binary expansion of the "Thue constant", 110110111110110111110110110..., wherein the 3nth bit is the complement of the nth.
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