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).

1, 3, 1, 15, 10, 3, 105, 105, 63, 15, 945, 1260, 1134, 540, 105, 10395, 17325, 20790, 14850, 5775, 945, 135135, 270270, 405405, 386100, 225225, 73710, 10395, 2027025, 4729725, 8513505, 10135125, 7882875, 3869775, 1091475, 135135, 34459425, 91891800, 192972780, 275675400, 268017750, 175429800, 74220300, 18378360, 2027025

Offset: 0

Author

Thomas Scheuerle, Jan 18 2025

Keywords

• nonn
• tabl

Examples

			Triangle begins:
n\k      0 |      1 |      2 |      3 |      4 |       5 |      6 |     7 |
[0]       1,
[1]       3,       1
[2]      15,      10,       3
[3]     105,     105,      63,       15
[4]     945,    1260,    1134,      540,     105
[5]   10395,   17325,   20790,    14850,    5775,     945
[6]  135135,  270270,  405405,   386100,  225225,   73710,   10395
[7] 2027025, 4729725, 8513505, 10135125, 7882875, 3869775, 1091475, 135135
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
• C. Nicholson, The probability integral for two variables, Biometrika 33 (1943), 59-72.
• Eric Weisstein's World of Mathematics, Owen T-Function.

Crossrefs

T(n, 1) = A001147(n+1), T(n, 2) = A000457(n-1), T(n, 3) = A001881(n+3)*3, T(n, n) = A001147(n).

Cf. A076729, (conj. row sums), A103327, A173424.

Programs

Maple

T := (n,k) -> 2^n*binomial(2*n + 1, 2*k + 1)*pochhammer(1/2, n - k)*pochhammer(1/2, k):
for n from 0 to 7 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Jan 21 2025

Mathematica

A380281[n_, k_] := (2*n - 1)!!*Binomial[n, k]*Binomial[2*n + 1, 2*k + 1]/Binomial[2*n, 2*k];
Table[A380281[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 22 2025 *)

PARI

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]

PARI

doublefact(n) = prod(i=0, (n-1)\2, n - 2*i )
T(n, k) = doublefact(2*n-1) * binomial(n, k) * binomial(2*n+1, 2*k+1) / binomial(2*n, 2*k)

SageMath

rf = rising_factorial
def T(n, k): return 2^n*binomial(2*n+1, 2*k+1)*rf(1/2, n-k)*rf(1/2, k)
for n in range(9): print([T(n, k) for k in range(n+1)])  # Peter Luschny, Jan 21 2025

Formula

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)!).

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).

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)!).

T(n, k) = A103327(n, k)*A173424(n, k).

T(n, k) = (2*n-1)!! * binomial(n, k) * binomial(2*n+1, 2*k+1) / binomial(2*n, 2*k).

Conjecture: Row sums are A076729.