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A157114
Triangle T(n, k) = binomial(n*k, n-k) + binomial(n*(n-k), k), read by rows.
4
2, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 56, 16, 1, 1, 25, 225, 225, 25, 1, 1, 36, 771, 1632, 771, 36, 1, 1, 49, 2597, 9261, 9261, 2597, 49, 1, 1, 64, 9136, 52384, 71920, 52384, 9136, 64, 1, 1, 81, 33777, 320814, 525987, 525987, 320814, 33777, 81, 1, 1, 100, 129130, 2090540, 4326015, 4237520, 4326015, 2090540, 129130, 100, 1
OFFSET
0,1
FORMULA
T(n, k) = binomial(n*k, n-k) + binomial(n*(n-k), k).
Sum_{k=0..n} T(n,k) = 2*A099237(n). - G. C. Greubel, Mar 09 2021
EXAMPLE
Triangle begins as:
2;
1, 1;
1, 4, 1;
1, 9, 9, 1;
1, 16, 56, 16, 1;
1, 25, 225, 225, 25, 1;
1, 36, 771, 1632, 771, 36, 1;
1, 49, 2597, 9261, 9261, 2597, 49, 1;
1, 64, 9136, 52384, 71920, 52384, 9136, 64, 1;
1, 81, 33777, 320814, 525987, 525987, 320814, 33777, 81, 1;
1, 100, 129130, 2090540, 4326015, 4237520, 4326015, 2090540, 129130, 100, 1;
MAPLE
A157114:= (n, k) -> binomial(n*k, n-k) + binomial(n*(n-k), k);
seq(seq(A157114(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 09 2021
MATHEMATICA
T[n_, k_]:= Binomial[n*k, n-k], Binomial[n*(n-k), k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Mar 09 2021 *)
PROG
(Sage)
def A157114(n, k): return binomial(n*k, n-k) + binomial(n*(n-k), k)
flatten([[A157114(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2021
(Magma)
A157114:= func< n, k | Binomial(n*k, n-k) + Binomial(n*(n-k), k) >;
[A157114(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 09 2021
CROSSREFS
Cf. A099237.
Sequence in context: A205552 A255707 A260757 * A156786 A156141 A174555
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 23 2009
EXTENSIONS
Edited by G. C. Greubel, Mar 09 2021
STATUS
approved