A173424 - OEIS

A173424

Triangle read by rows: T(n, k) = (2*n - 2*k)!*(2*k)!/(2^n*(n - k)!*k!).

1, 1, 1, 3, 1, 3, 15, 3, 3, 15, 105, 15, 9, 15, 105, 945, 105, 45, 45, 105, 945, 10395, 945, 315, 225, 315, 945, 10395, 135135, 10395, 2835, 1575, 1575, 2835, 10395, 135135, 2027025, 135135, 31185, 14175, 11025, 14175, 31185, 135135, 2027025, 34459425

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OFFSET

0,4

LINKS

Table of n, a(n) for n=0..45.

FORMULA

T(n, k) = A006882(2*n-2*k-1) * A006882(2*k-1).

T(n, k) = A001147(n-k) * A001147(k).

From Peter Luschny, Apr 15 2023: (Start)

T(n, k) = (1/Pi) * 2^n * Gamma(k + 1/2) * Gamma(n - k + 1/2).

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

EXAMPLE

Triangle T(n, k) starts:

[0] 1;

[1] 1, 1;

[2] 3, 1, 3;

[3] 15, 3, 3, 15;

[4] 105, 15, 9, 15, 105;

[5] 945, 105, 45, 45, 105, 945;

[6] 10395, 945, 315, 225, 315, 945, 10395;

[7] 135135, 10395, 2835, 1575, 1575, 2835, 10395, 135135;

[8] 2027025, 135135, 31185, 14175, 11025, 14175, 31185, 135135, 2027025;

MAPLE

T := (n, k) -> doublefactorial(2*n-1) * binomial(n, k) / binomial(2*n, 2*k):

for n from 0 to 8 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Apr 15 2023

MATHEMATICA

t[n_, k_] = (2*n - 2*k)!*(2*k)!/(2^n*(n - k)!*k!);

Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}]; Flatten[%]

CROSSREFS

Cf. A034430 (row sums), A006882, A001147.

Sequence in context: A351372 A356411 A355793 * A143081 A179658 A112811

Adjacent sequences: A173421 A173422 A173423 * A173425 A173426 A173427

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Feb 18 2010

EXTENSIONS

Formula added by the Assoc. Editors of the OEIS, Feb 24 2010

STATUS

approved