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A157219
Triangle T(n, k) = binomial(n*f(n,k), f(n,k)), where f(n, k) = k if k <= floor(n/2) otherwise n-k, read by rows.
3
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 28, 4, 1, 1, 5, 45, 45, 5, 1, 1, 6, 66, 816, 66, 6, 1, 1, 7, 91, 1330, 1330, 91, 7, 1, 1, 8, 120, 2024, 35960, 2024, 120, 8, 1, 1, 9, 153, 2925, 58905, 58905, 2925, 153, 9, 1, 1, 10, 190, 4060, 91390, 2118760, 91390, 4060, 190, 10, 1
OFFSET
0,5
FORMULA
T(n, k) = binomial(n*f(n,k), f(n,k)), where f(n, k) = k if k <= floor(n/2) otherwise n-k.
T(n, n-k) = T(n, k).
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 4, 28, 4, 1;
1, 5, 45, 45, 5, 1;
1, 6, 66, 816, 66, 6, 1;
1, 7, 91, 1330, 1330, 91, 7, 1;
1, 8, 120, 2024, 35960, 2024, 120, 8, 1;
1, 9, 153, 2925, 58905, 58905, 2925, 153, 9, 1;
1, 10, 190, 4060, 91390, 2118760, 91390, 4060, 190, 10, 1;
MATHEMATICA
f[n_, k_]:= If[k<=Floor[n/2], k, n-k];
T[n_, k_]:= Binomial[n*f[n, k], f[n, k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
PROG
(Magma)
f:= func< n, k | k le Floor(n/2) select k else n-k >;
[Binomial(n*f(n, k), f(n, k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 10 2022
(Sage)
def f(n, k): return k if (k <= (n//2)) else n-k
flatten([[binomial(n*f(n, k), f(n, k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 10 2022
CROSSREFS
Sequence in context: A159623 A143199 A137896 * A167040 A054450 A344610
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 25 2009
EXTENSIONS
Edited by G. C. Greubel, Jan 10 2022
STATUS
approved