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A144403
Triangle T(n, k) = binomial(n, k)^2 - binomial(n, k) - 1, read by rows.
1
-1, -1, -1, -1, 1, -1, -1, 5, 5, -1, -1, 11, 29, 11, -1, -1, 19, 89, 89, 19, -1, -1, 29, 209, 379, 209, 29, -1, -1, 41, 419, 1189, 1189, 419, 41, -1, -1, 55, 755, 3079, 4829, 3079, 755, 55, -1, -1, 71, 1259, 6971, 15749, 15749, 6971, 1259, 71, -1, -1, 89, 1979, 14279, 43889, 63251, 43889, 14279, 1979, 89, -1
OFFSET
0,8
FORMULA
T(n, k) = binomial(n, k)^2 - binomial(n, k) - 1.
Sum_{k=0..n} T(n,k) = Binomial(2*n, n) - 2^n - n - 1. - G. C. Greubel, Mar 27 2021
EXAMPLE
Triangle begins as:
-1;
-1, -1;
-1, 1, -1;
-1, 5, 5, -1;
-1, 11, 29, 11, -1;
-1, 19, 89, 89, 19, -1;
-1, 29, 209, 379, 209, 29, -1;
-1, 41, 419, 1189, 1189, 419, 41, -1;
-1, 55, 755, 3079, 4829, 3079, 755, 55, -1;
-1, 71, 1259, 6971, 15749, 15749, 6971, 1259, 71, -1;
-1, 89, 1979, 14279, 43889, 63251, 43889, 14279, 1979, 89, -1;
MAPLE
A144403:= (n, k)-> binomial(n, k)^2 - binomial(n, k) - 1;
seq(seq(A144403(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 27 2021
MATHEMATICA
Table[Binomial[n, m]^2 -Binomial[n, m] -1, {n, 0, 12}, {m, 0, n}]//Flatten
PROG
(Magma) [Binomial(n, k)^2 - Binomial(n, k) - 1: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 27 2021
(Sage) flatten([[binomial(n, k)^2 - binomial(n, k) - 1 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 27 2021
CROSSREFS
Cf. A000984.
Sequence in context: A046568 A046571 A172349 * A188587 A373434 A174119
KEYWORD
sign,tabl
AUTHOR
EXTENSIONS
Edited by G. C. Greubel, Mar 27 2021
STATUS
approved