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%I #6 Mar 27 2021 23:52:25
%S 1,1,1,1,18,1,1,69,69,1,1,172,606,172,1,1,345,2890,2890,345,1,1,606,
%T 9885,23580,9885,606,1,1,973,27321,127365,127365,27321,973,1,1,1464,
%U 65044,523656,1024030,523656,65044,1464,1,1,2097,138636,1770972,5985126,5985126,1770972,138636,2097,1
%N Triangle T(n,k) = binomial(n, k)*(3*binomial(n, k)^2 - binomial(n, k) - 1), read by rows.
%H G. C. Greubel, <a href="/A144405/b144405.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n,k) = binomial(n, k)*(3*binomial(n, k)^2 - binomial(n, k) - 1).
%F 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
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 18, 1;
%e 1, 69, 69, 1;
%e 1, 172, 606, 172, 1;
%e 1, 345, 2890, 2890, 345, 1;
%e 1, 606, 9885, 23580, 9885, 606, 1;
%e 1, 973, 27321, 127365, 127365, 27321, 973, 1;
%e 1, 1464, 65044, 523656, 1024030, 523656, 65044, 1464, 1;
%e 1, 2097, 138636, 1770972, 5985126, 5985126, 1770972, 138636, 2097, 1;
%e 1, 2890, 271305, 5169480, 27738690, 47945268, 27738690, 5169480, 271305, 2890, 1;
%p A144405:= (n,k) -> binomial(n, k)*(3*binomial(n, k)^2 - binomial(n, k) - 1);
%p seq(seq( A144405(n,k), k=0..n), n=0..12); # _G. C. Greubel_, Mar 27 2021
%t Table[Table[Binomial[n, m]*(3*Binomial[n, m]^2 - Binomial[n, m] - 1), {m, 0, n}], {n, 0, 10}]; Flatten[%]
%o (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
%o (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
%Y Cf. A000172, A000984, A144404.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 03 2008
%E Edited by _G. C. Greubel_, Mar 27 2021
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