A246257 Triangular array read by rows: T(n, k) = S(n, [n/2]-k) and S(n,k) = C(n, 2*k)*(2*k-1)!!*((2*k-1)!! + 1)/2, n>=0, 0<=k<=[n/2].

1, 1, 1, 1, 3, 1, 6, 6, 1, 30, 10, 1, 120, 90, 15, 1, 840, 210, 21, 1, 5565, 3360, 420, 28, 1, 50085, 10080, 756, 36, 1, 446985, 250425, 25200, 1260, 45, 1, 4916835, 918225, 55440, 1980, 55, 1, 54033210, 29501010, 2754675, 110880, 2970, 66, 1

Offset: 0

Author

Peter Luschny, Aug 21 2014

Keywords

• tabf
• nonn

Examples

			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.
		

Crossrefs

Cf. A002771 (row sums), A246256, A096713.

Programs

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 *)

Sage

from sage.functions.hypergeometric import closed_form
def A246257_row(n):
    R. = ZZ[]
    h = hypergeometric([-n/2,(1-n)/2], [], 2*z)
    g = hypergeometric([1/2,-n/2,(1-n)/2], [], 4*z)
    T = R(((closed_form(h)+closed_form(g))/2)).coefficients()
    return T[::-1]
for n in range(13): A246257_row(n)

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.