OFFSET
0,2
FORMULA
Special values of generalized hypergeometric functions of type 5F4, in Maple notation: S(n,k) = (((-1)^k*(3^(-n))*k/(4*(k!))) *(-12*GAMMA(3*n)*hypergeom([1/3-k/3,2/3-k/3,1-k/3,n+1/3,n+2/3],[1/3,2/3,2/3,4/3],1)/GAMMA(n)+6*(k-1)*GAMMA(4/3)*GAMMA(1+3*n)*hypergeom([2/3-k/3,1-k/3,4/3-k/3,2/3+n,n+1],[2/3,1,4/3,5/3],1)/GAMMA(n+1/3)-(k-2)*(k-1)*GAMMA(5/3)*GAMMA(3*n+2)*hypergeom([1-k/3,4/3-k/3,5/3-k/3,n+1,n+4/3],[4/3,4/3,5/3,2],1)/GAMMA(n+2/3)))
EXAMPLE
Example: S(n,k) in table form for n=0..4;
1
2,1
40,50,14,1
2240, 4240, 2200, 440, 36, 1
246400, 608960, 447200, 141520, 22080, 1760, 68, 1.
MAPLE
S:=proc(n, k) (((-1)^k*(3^(-n))*k/(4*(k!))) *(-12*GAMMA(3*n)*hypergeom([1/3-k/3, 2/3-k/3, 1-k/3, n+1/3, n+2/3], [1/3, 2/3, 2/3, 4/3], 1)/GAMMA(n)+6*(k-1)*GAMMA(4/3)*GAMMA(1+3*n)*hypergeom([2/3-k/3, 1-k/3, 4/3-k/3, 2/3+n, n+1], [2/3, 1, 4/3, 5/3], 1)/GAMMA(n+1/3)-(k-2)*(k-1)*GAMMA(5/3)*GAMMA(3*n+2)*hypergeom([1-k/3, 4/3-k/3, 5/3-k/3, n+1, n+4/3], [4/3, 4/3, 5/3, 2], 1)/GAMMA(n+2/3))); end;
for n from 1 to 6 do seq(round(evalf(S(n, kk))), kk=1..2*n) end do;
# The above Maple program reproduces the data without the initial value 1.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Karol A. Penson, Apr 01 2016
STATUS
approved