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A131067 Triangle read by rows: T(n,k) = 7*binomial(n,k) - 6 for 0 <= k <= n.

%I A131067 #12 Sep 08 2022 08:45:30
%S A131067 1,1,1,1,8,1,1,15,15,1,1,22,36,22,1,1,29,64,64,29,1,1,36,99,134,99,36,
%T A131067 1,1,43,141,239,239,141,43,1,1,50,190,386,484,386,190,50,1,1,57,246,
%U A131067 582,876,876,582,246,57,1,1,64,309,834,1464,1758,1464,834,309,64,1
%N A131067 Triangle read by rows: T(n,k) = 7*binomial(n,k) - 6 for 0 <= k <= n.
%C A131067 Row sums = A131068: (1, 2, 10, 32, 82, 188, 406, ...), the binomial transform of (1, 1, 7, 7, 7, ...).
%H A131067 G. C. Greubel, <a href="/A131067/b131067.txt">Rows n = 0..100 of triangle, flattened</a>
%F A131067 G.f.: G(t,z) = (1-z-t*z+7*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - _Emeric Deutsch_, Jun 20 2007
%e A131067 First few rows of the triangle:
%e A131067   1;
%e A131067   1,  1;
%e A131067   1,  8,  1;
%e A131067   1, 15, 15,  1;
%e A131067   1, 22, 36, 22,  1;
%e A131067   1, 29, 64, 64, 29, 1;
%e A131067   ...
%p A131067 T := proc (n, k) if k <= n then 7*binomial(n, k)-6 else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # _Emeric Deutsch_, Jun 20 2007
%t A131067 Table[7*Binomial[n, k] -6, {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 12 2020 *)
%o A131067 (Magma) [7*Binomial(n, k) -6: k in [0..n], n in [0..10]]; // _G. C. Greubel_, Mar 12 2020
%o A131067 (Sage) [[7*binomial(n, k) -6 for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Mar 12 2020
%Y A131067 Cf. A131064, A131066.
%Y A131067 Sequence m*binomial(n,k) - (m-1): A007318 (m=1), A109128 (m=2), A131060 (m=3), A131061 (m=4), A131063 (m=5), A131065 (m=6), this sequence (m=7), A131068 (m=8).
%K A131067 nonn,tabl
%O A131067 0,5
%A A131067 _Gary W. Adamson_, Jun 13 2007
%E A131067 More terms from _Emeric Deutsch_, Jun 20 2007