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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].

%I A246257 #10 Aug 02 2019 12:56:57
%S A246257 1,1,1,1,3,1,6,6,1,30,10,1,120,90,15,1,840,210,21,1,5565,3360,420,28,
%T A246257 1,50085,10080,756,36,1,446985,250425,25200,1260,45,1,4916835,918225,
%U A246257 55440,1980,55,1,54033210,29501010,2754675,110880,2970,66,1
%N 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].
%F A246257 T(n, k) = (n!/(j!*(n-2*j)!))*(2^(-j-1)+Gamma(j+1/2)/sqrt(4*Pi)) where j = floor(n/2) - k.
%e A246257 Triangle starts:
%e A246257 [ 0] 1,
%e A246257 [ 1] 1,
%e A246257 [ 2] 1, 1,
%e A246257 [ 3] 3, 1,
%e A246257 [ 4] 6, 6, 1,
%e A246257 [ 5] 30, 10, 1,
%e A246257 [ 6] 120, 90, 15, 1,
%e A246257 [ 7] 840, 210, 21, 1,
%e A246257 [ 8] 5565, 3360, 420, 28, 1,
%e A246257 [ 9] 50085, 10080, 756, 36, 1,
%e A246257 [10] 446985, 250425, 25200, 1260, 45, 1.
%p A246257 T := proc(n, k) local j; j := iquo(n,2) - k;
%p A246257 (n!/(j!*(n-2*j)!))*(2^(-j-1)+GAMMA(j+1/2)/sqrt(4*Pi)) end:
%p A246257 seq(print(seq(T(n,k), k=0..iquo(n,2))), n=0..10);
%t A246257 row[n_] := FunctionExpand[HypergeometricPFQ[{-n/2, (1-n)/2}, {}, 2z] + HypergeometricPFQ[{1/2, -n/2, (1-n)/2}, {}, 4z]]/2 // CoefficientList[#, z]& // Reverse;
%t A246257 Table[row[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Aug 02 2019 *)
%o A246257 (Sage)
%o A246257 from sage.functions.hypergeometric import closed_form
%o A246257 def A246257_row(n):
%o A246257     R.<z> = ZZ[]
%o A246257     h = hypergeometric([-n/2,(1-n)/2], [], 2*z)
%o A246257     g = hypergeometric([1/2,-n/2,(1-n)/2], [], 4*z)
%o A246257     T = R(((closed_form(h)+closed_form(g))/2)).coefficients()
%o A246257     return T[::-1]
%o A246257 for n in range(13): A246257_row(n)
%Y A246257 Cf. A002771 (row sums), A246256, A096713.
%K A246257 tabf,nonn
%O A246257 0,5
%A A246257 _Peter Luschny_, Aug 21 2014