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A132339
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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.
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6
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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
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refs;
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OFFSET
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0,5
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LINKS
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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.
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FORMULA
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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).
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EXAMPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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sign,tabl,easy
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AUTHOR
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N. J. A. Sloane, Nov 08 2007
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EXTENSIONS
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More terms from Max Alekseyev, Sep 12 2009
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STATUS
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approved
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