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A176567
Triangle T(n, k) = binomial(binomial(n, 2) + k, k) + binomial(binomial(n, 2) + n-k, n-k) - binomial(binomial(n, 2) + n, n), read by rows.
2
1, 1, 1, 1, 1, 1, 1, -6, -6, 1, 1, -119, -154, -119, 1, 1, -1991, -2651, -2651, -1991, 1, 1, -38744, -50252, -52632, -50252, -38744, 1, 1, -888008, -1118007, -1169366, -1169366, -1118007, -888008, 1, 1, -23535791, -28915001, -30018509, -30188420, -30018509, -28915001, -23535791, 1
OFFSET
0,8
FORMULA
T(n, k) = binomial(binomial(n, 2) + k, k) + binomial(binomial(n, 2) + n-k, n-k) - binomial(binomial(n, 2) + n, n).
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 1, 1;
1, -6, -6, 1;
1, -119, -154, -119, 1;
1, -1991, -2651, -2651, -1991, 1;
1, -38744, -50252, -52632, -50252, -38744, 1;
1, -888008, -1118007, -1169366, -1169366, -1118007, -888008, 1;
MATHEMATICA
T[n_, k_]= Binomial[Binomial[n, 2] + k, k] + Binomial[Binomial[n, 2] + n-k, n-k] - Binomial[Binomial[n, 2] + n, n];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
PROG
(Magma)
T:= func< n, k | Binomial(Binomial(n, 2) +k, k) + Binomial(Binomial(n, 2) +n-k, n-k) - Binomial(Binomial(n, 2) +n, n) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2021
(Sage)
def f(n, k): return binomial(binomial(n, 2) + k, k)
def T(n, k): return f(n, k) + f(n, n-k) - f(n, n)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 09 2021
CROSSREFS
Sequence in context: A155868 A322622 A176565 * A372272 A283100 A065493
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Apr 22 2010
EXTENSIONS
Edited by G. C. Greubel, Jul 09 2021
STATUS
approved