VOOZH about

URL: https://oeis.org/A132339

⇱ A132339 - OEIS

login
A132339
Array T(n, k) = (-1)^(n+k)*(n+k-2)!*(2*n+2*k-2)!/(n!*k!*(2*n-1)!*(2*k-1)!), with T(0, 0) = 1, T(0, 1) = T(1, 0) = -1, read by antidiagonals.
6
1, -1, -1, 0, 2, 0, 0, -2, -2, 0, 0, 2, 10, 2, 0, 0, -2, -28, -28, -2, 0, 0, 2, 60, 168, 60, 2, 0, 0, -2, -110, -660, -660, -110, -2, 0, 0, 2, 182, 2002, 4290, 2002, 182, 2, 0, 0, -2, -280, -5096, -20020, -20020, -5096, -280, -2, 0, 0, 2, 408, 11424, 74256, 136136, 74256, 11424, 408, 2, 0
OFFSET
0,5
LINKS
G. Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, 6 (1965), circa p. 82.
FORMULA
T(n, k) = (-1)^(n+k)*(n+k-2)!*(2*n+2*k-2)!/(n!*k!*(2*n-1)!*(2*k-1)!), with T(0, 0) = 1, T(0, 1) = T(1, 0) = -1.
A(n, k) = T(n-k, k) (antidiagonals).
A(n, n-k) = A(n, k).
A(2*n, n) = A132341(n).
EXAMPLE
Array (T(n,k)) begins:
1, -1, 0, 0, 0, 0, 0 ... A154955(k)
-1, 2, -2, 2, -2, 2, -2 ... (-1)^(k+1)*A040000(k)
0, -2, 10, -28, 60, -110, 182 ... (-1)^k*A006331(k)
0, 2, -28, 168, -660, 2002, -5096 ... (-1)^k*A006332(k)
0, -2, 60, -660, 4290, -20020, 74256 ... (-1)^k*A006333(k)
0, 2, -110, 2002, -20020, 136136, -705432 ... (-1)^k*A006334(k)
0, -2, 182, -5096, 74256, -705432, 4938024 ...
0, 2, -280, 11424, -232560, 2984520, -27457584 ...
Antidiagonal (A(n,k)) triangle begins as:
1;
-1, -1;
0, 2, 0;
0, -2, -2, 0;
0, 2, 10, 2, 0;
0, -2, -28, -28, -2, 0;
0, 2, 60, 168, 60, 2, 0;
0, -2, -110, -660, -660, -110, -2, 0;
0, 2, 182, 2002, 4290, 2002, 182, 2, 0;
0, -2, -280, -5096, -20020, -20020, -5096, -280, -2, 0;
0, 2, 408, 11424, 74256, 136136, 74256, 11424, 408, 2, 0;
MATHEMATICA
Flatten[{{1}, {-1, -1}}~Join~Table[(2(-1)^(#+k)*(#+k-1)!*(2#+2k-3)!)/(#!*k!*(2# - 1)!*(2k-1)!) &@(n-k), {n, 2, 12}, {k, 0, n}]] (* Michael De Vlieger, Mar 26 2016 *)
PROG
(SageMath)
f=factorial
def T(n, k):
if (k==0): return bool(n==0) - bool(n==1)
elif (n==0): return bool(k==0) - bool(k==1)
else: return (-1)^(n+k)*f(n+k-2)*f(2*n+2*k-2)/(f(n)*f(k)*f(2*n-1)*f(2*k-1))
flatten([[T(n-k, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Dec 14 2021
KEYWORD
sign,tabl,easy
AUTHOR
N. J. A. Sloane, Nov 08 2007
EXTENSIONS
More terms from Max Alekseyev, Sep 12 2009
STATUS
approved