%I A087659
%S A087659 1,6,57,701,10147,164317,2888282,54047434,1062530119,21739192762,
%T A087659 459685114665,9993072855135,222421656113435,5052215132332492,
%U A087659 116808526607319823,2742986603349411311,65306671610636210891
%N A087659 Values of a certain hypergeometric function: a(n) = hypergeom([ -n, (n+4)/
               2, (n+5)/2],[3, 2],-4).
%C A087659 Bill Gosper, Feb 04 2004: "A few weeks ago I conjectured that
%C A087659 "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).
%C A087659 "This is the special case C(3,i,n-i) of C(m,k,n) :=
%C A087659 "(n+k)!(n+m)!/(n!(n+m+k)!) * Product_{j=1..k} (j - 1)! (n + j m + m)!/
               ((m + j - 1)! (n + j m)!)
%C A087659 "which I also conjecture integral."
%C A087659 Alec Mihailovs, Feb 04 2004: "These conjectures are true. Consider the 
               partition
%C A087659 "p(m,k,n)=(n+m,m,...,m) of n+m*(k+1), where m is repeated k times. It 
               is easy
%C A087659 "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.
%C A087659 "Another formula for C(m,k,n) is ((n+mk+m)!/n!) * Product_{i=0..m-1} 
               i!/((k+i)!(n+k+i+1)!)."
%C A087659 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.
%F A087659 Also equals Sum _{i=0..n} 2 C(n, i) (n + 2 i + 3)! / ( (i + 1)! (i + 
               2)! (n + 3)! ).
%o A087659 (PARI) a(n)= sum(i=0,n,2*binomial(n,i)*(n+2*i+3)!/((i+1)!*(i+2)!*(n+3)!)) 
               (from Benoit Cloitre)
%Y A087659 Row sums of triangle A087727. Cf. A087660-A087662.
%Y A087659 Sequence in context: A153851 A141372 A152170 this_sequence A107718 A000406 
               A032119
%Y A087659 Adjacent sequences: A087656 A087657 A087658 this_sequence A087660 A087661 
               A087662
%K A087659 nonn
%O A087659 0,2
%A A087659 R. William Gosper (rwg(AT)tc.spnet.com), Sep 26 2003
%E A087659 More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 26 2003