OFFSET
0,5
FORMULA
T(n, k) = (n!/(j!*(n-2*j)!))*(2^(-j-1)+Gamma(j+1/2)/sqrt(4*Pi)) where j = floor(n/2) - k.
EXAMPLE
Triangle starts:
[ 0] 1,
[ 1] 1,
[ 2] 1, 1,
[ 3] 3, 1,
[ 4] 6, 6, 1,
[ 5] 30, 10, 1,
[ 6] 120, 90, 15, 1,
[ 7] 840, 210, 21, 1,
[ 8] 5565, 3360, 420, 28, 1,
[ 9] 50085, 10080, 756, 36, 1,
[10] 446985, 250425, 25200, 1260, 45, 1.
MAPLE
T := proc(n, k) local j; j := iquo(n, 2) - k;
(n!/(j!*(n-2*j)!))*(2^(-j-1)+GAMMA(j+1/2)/sqrt(4*Pi)) end:
seq(print(seq(T(n, k), k=0..iquo(n, 2))), n=0..10);
MATHEMATICA
row[n_] := FunctionExpand[HypergeometricPFQ[{-n/2, (1-n)/2}, {}, 2z] + HypergeometricPFQ[{1/2, -n/2, (1-n)/2}, {}, 4z]]/2 // CoefficientList[#, z]& // Reverse;
Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 02 2019 *)
PROG
CROSSREFS
KEYWORD
tabf,nonn
AUTHOR
Peter Luschny, Aug 21 2014
STATUS
approved