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A380281 Triangle T(n, k) read by rows: T(n, k) = 2^n*binomial(2*n + 1, 2*k + 1) * Pochhammer(1/2, n - k) * Pochhammer(1/2, k).

%I A380281 #48 Jan 22 2025 11:34:13
%S A380281 1,3,1,15,10,3,105,105,63,15,945,1260,1134,540,105,10395,17325,20790,
%T A380281 14850,5775,945,135135,270270,405405,386100,225225,73710,10395,
%U A380281 2027025,4729725,8513505,10135125,7882875,3869775,1091475,135135,34459425,91891800,192972780,275675400,268017750,175429800,74220300,18378360,2027025
%N A380281 Triangle T(n, k) read by rows: T(n, k) = 2^n*binomial(2*n + 1, 2*k + 1) * Pochhammer(1/2, n - k) * Pochhammer(1/2, k).
%H A380281 Paolo Xausa, <a href="/A380281/b380281.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of triangle, flattened).
%H A380281 C. Nicholson, <a href="https://www.jstor.org/stable/2333620">The probability integral for two variables</a>, Biometrika 33 (1943), 59-72.
%H A380281 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OwenT-Function.html">Owen T-Function</a>.
%F A380281 Coefficients for the series representation of Owen's T-function Ot(x, m) = atan(m)/(2*Pi) + Sum_{s>=0} (-1)^(s+1)*m*(Sum_{r=0..s} T(s, r)*m^(2*s))*x^(2+2*s)/(2*Pi*(2+2*s)!).
%F A380281 Ot(x, m) - atan(m)/(2*Pi) = -V(x, x*m), where V is Nicholson's V-function. V(h, q) = Integral_{x=0..h} Integral_{y=0..q*x/h} phi(x)*phi(y) dydx, where phi(x) is the standard normal density exp(-x^2/2)/sqrt(2*Pi).
%F A380281 G.f. of row n: ((1 + x)^(n+1)*Hypergeometric2F1[1, 1/2 + n + 1, 3/2, -x]*(2*(n+1))!)/(2^(n+1)*(n+1)!).
%F A380281 T(n, k) = A103327(n, k)*A173424(n, k).
%F A380281 T(n, k) = (2*n-1)!! * binomial(n, k) * binomial(2*n+1, 2*k+1) / binomial(2*n, 2*k).
%F A380281 Conjecture: Row sums are A076729.
%e A380281 Triangle begins:
%e A380281 n\k      0 |      1 |      2 |      3 |      4 |       5 |      6 |     7 |
%e A380281 [0]       1,
%e A380281 [1]       3,       1
%e A380281 [2]      15,      10,       3
%e A380281 [3]     105,     105,      63,       15
%e A380281 [4]     945,    1260,    1134,      540,     105
%e A380281 [5]   10395,   17325,   20790,    14850,    5775,     945
%e A380281 [6]  135135,  270270,  405405,   386100,  225225,   73710,   10395
%e A380281 [7] 2027025, 4729725, 8513505, 10135125, 7882875, 3869775, 1091475, 135135
%p A380281 T := (n,k) -> 2^n*binomial(2*n + 1, 2*k + 1)*pochhammer(1/2, n - k)*pochhammer(1/2, k):
%p A380281 for n from 0 to 7 do seq(T(n, k), k = 0..n) od;  # _Peter Luschny_, Jan 21 2025
%t A380281 A380281[n_, k_] := (2*n - 1)!!*Binomial[n, k]*Binomial[2*n + 1, 2*k + 1]/Binomial[2*n, 2*k];
%t A380281 Table[A380281[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Jan 22 2025 *)
%o A380281 (PARI)
%o A380281 T(n, k) = Vec(O(x^(1+n))+(1+x)^(n+1)*hypergeom([1,1/2+n+1],3/2,-x)*(2*(n+1))!/(2^(n+1)*(n+1)!))[1+k]
%o A380281 (PARI)
%o A380281 doublefact(n) = prod(i=0, (n-1)\2, n - 2*i )
%o A380281 T(n, k) = doublefact(2*n-1) * binomial(n, k) * binomial(2*n+1, 2*k+1) / binomial(2*n, 2*k)
%o A380281 (SageMath)
%o A380281 rf = rising_factorial
%o A380281 def T(n, k): return 2^n*binomial(2*n+1, 2*k+1)*rf(1/2, n-k)*rf(1/2, k)
%o A380281 for n in range(9): print([T(n, k) for k in range(n+1)])  # _Peter Luschny_, Jan 21 2025
%Y A380281 T(n, 1) = A001147(n+1), T(n, 2) = A000457(n-1), T(n, 3) = A001881(n+3)*3, T(n, n) = A001147(n).
%Y A380281 Cf. A076729, (conj. row sums), A103327, A173424.
%K A380281 nonn,tabl
%O A380281 0,2
%A A380281 _Thomas Scheuerle_, Jan 18 2025