A144405 Triangle T(n,k) = binomial(n, k)*(3*binomial(n, k)^2 - binomial(n, k) - 1), read by rows.
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
Examples
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;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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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
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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
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Mathematica
Table[Table[Binomial[n, m]*(3*Binomial[n, m]^2 - Binomial[n, m] - 1), {m, 0, n}], {n, 0, 10}]; Flatten[%] -
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
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
Extensions
Edited by G. C. Greubel, Mar 27 2021