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A144405
Triangle T(n,k) = binomial(n, k)*(3*binomial(n, k)^2 - binomial(n, k) - 1), read by rows.
2
1, 1, 1, 1, 18, 1, 1, 69, 69, 1, 1, 172, 606, 172, 1, 1, 345, 2890, 2890, 345, 1, 1, 606, 9885, 23580, 9885, 606, 1, 1, 973, 27321, 127365, 127365, 27321, 973, 1, 1, 1464, 65044, 523656, 1024030, 523656, 65044, 1464, 1, 1, 2097, 138636, 1770972, 5985126, 5985126, 1770972, 138636, 2097, 1
OFFSET
0,5
FORMULA
T(n,k) = binomial(n, k)*(3*binomial(n, k)^2 - binomial(n, k) - 1).
Sum_{k=0..n} T(n, k) = A000172(n) - A000984(n) - 2^n = Hypergeometric3F2([-n, -n, -n], [1, 1], -1) - binomial(2*n, n) - 2^n. - G. C. Greubel, Mar 27 2021
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 18, 1;
1, 69, 69, 1;
1, 172, 606, 172, 1;
1, 345, 2890, 2890, 345, 1;
1, 606, 9885, 23580, 9885, 606, 1;
1, 973, 27321, 127365, 127365, 27321, 973, 1;
1, 1464, 65044, 523656, 1024030, 523656, 65044, 1464, 1;
1, 2097, 138636, 1770972, 5985126, 5985126, 1770972, 138636, 2097, 1;
1, 2890, 271305, 5169480, 27738690, 47945268, 27738690, 5169480, 271305, 2890, 1;
MAPLE
A144405:= (n, k) -> binomial(n, k)*(3*binomial(n, k)^2 - binomial(n, k) - 1);
seq(seq( A144405(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 27 2021
MATHEMATICA
Table[Table[Binomial[n, m]*(3*Binomial[n, m]^2 - Binomial[n, m] - 1), {m, 0, n}], {n, 0, 10}]; Flatten[%]
PROG
(Magma) [Binomial(n, k)*(3*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)*(3*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
KEYWORD
nonn,tabl
AUTHOR
EXTENSIONS
Edited by G. C. Greubel, Mar 27 2021
STATUS
approved