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.

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. C. Greubel, Antidiagonals n = 0..50, flattened

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

(Sage)
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

Crossrefs

Cf. A006331, A006332, A006333, A006334, A040000, A132341, A154955.

Sequence in context: A326915 A099766 A194947 * A333941 A137676 A333755

Adjacent sequences: A132336 A132337 A132338 * A132340 A132341 A132342

Keyword

sign,tabl,easy,changed

Author

N. J. A. Sloane, Nov 08 2007

Extensions

More terms from Max Alekseyev, Sep 12 2009

Status

approved