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A246256 Triangular array read by rows. Row n lists the coefficients of the closed form of hypergeometric([1/2, -n/2, (1-n)/2], [], 4*z).

%I A246256 #17 May 05 2024 05:35:38
%S A246256 1,1,1,1,3,1,9,6,1,45,10,1,225,135,15,1,1575,315,21,1,11025,6300,630,
%T A246256 28,1,99225,18900,1134,36,1,893025,496125,47250,1890,45,1,9823275,
%U A246256 1819125,103950,2970,55,1,108056025,58939650,5457375,207900,4455,66,1
%N A246256 Triangular array read by rows. Row n lists the coefficients of the closed form of hypergeometric([1/2, -n/2, (1-n)/2], [], 4*z).
%F A246256 For the e.g.f. and a recurrence see the Maple program.
%F A246256 T(n, k) = (Product_{j=0..(floor(n/2) - k)} (2*j - 1)^2)*binomial(n, 2*k + (n mod 2)). - _Detlef Meya_, May 05 2024
%e A246256 Triangle starts:
%e A246256 [ 0] 1,
%e A246256 [ 1] 1,
%e A246256 [ 2] 1, 1,
%e A246256 [ 3] 3, 1,
%e A246256 [ 4] 9, 6, 1,
%e A246256 [ 5] 45, 10, 1,
%e A246256 [ 6] 225, 135, 15, 1,
%e A246256 [ 7] 1575, 315, 21, 1,
%e A246256 [ 8] 11025, 6300, 630, 28, 1,
%e A246256 [ 9] 99225, 18900, 1134, 36, 1,
%e A246256 [10] 893025, 496125, 47250, 1890, 45, 1,
%e A246256 [11] 9823275, 1819125, 103950, 2970, 55, 1,
%e A246256 ...
%p A246256 g := exp(x*z)/sqrt((1-z)/(1+z)); gser := n -> series(g, z, n+2):
%p A246256 seq(seq(coeff(n!*coeff(gser(n),z,n),x,2*i+irem(n,2)),i=0..iquo(n,2)),n=0..12);
%p A246256 # Recurrence for A138022 from Robert Israel.
%p A246256 T := proc(n, k) option remember;
%p A246256 if k < 0 or n < k then 0 elif k = n then 1 elif k = n-1 then n
%p A246256 elif k = 0 then T(n-1,k)+(n-2)*(n-1)*T(n-2,k)
%p A246256 else T(n-1,k)+T(n-1,k-1)+(n-2)*(n-1)*(T(n-2,k)-T(n-3,k-1)) fi end:
%p A246256 A246256_row := n -> seq(T(n,2*k+(n mod 2)),k=0..iquo(n,2)):
%p A246256 seq(A246256_row(n), n=0..12);
%t A246256 row[n_] := HypergeometricPFQ[{1/2, -n/2, (1-n)/2}, {}, 4z] // FunctionExpand // CoefficientList[#, z]& // Reverse;
%t A246256 Table[row[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Aug 02 2019 *)
%t A246256 T[n_, k_] := Product[(2*j - 1)^2, {j, 0, Floor[n/2] - k}]*Binomial[n, 2*k + Mod[n,2]]; Flatten[Table[T[n,k],{n, 0, 12},{k, 0 ,Floor[n/2]}]] (*  _Detlef Meya_, May 05 2024 *)
%o A246256 (Sage)
%o A246256 from sage.functions.hypergeometric import closed_form
%o A246256 def A246256_row(n):
%o A246256     R.<z> = ZZ[]
%o A246256     h = hypergeometric([1/2,-n/2,(1-n)/2], [], 4*z)
%o A246256     T = R(closed_form(h)).coefficients()
%o A246256     return T[::-1]
%o A246256 for n in range(13): A246256_row(n)
%Y A246256 Cf. A081919 (row sums), A138022, A246257.
%K A246256 tabf,nonn
%O A246256 0,5
%A A246256 _Peter Luschny_, Aug 21 2014