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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

G. C. Greubel, Antidiagnals 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

AUTHOR

N. J. A. Sloane, Nov 08 2007

EXTENSIONS

More terms from Max Alekseyev, Sep 12 2009

STATUS

approved

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Last modified December 10 02:09 EST 2022. Contains 358712 sequences. (Running on oeis4.)