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A144405 Triangle T(n,k) = binomial(n, k)*(3*binomial(n, k)^2 - binomial(n, k) - 1), read by rows.

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