A144405 - OEIS

%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