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A173045 Triangle T(n, k, q) = binomial(n, k) - 1 + q^(n*binomial(n-2, k-1)) with T(n, 0, q) = T(n, n, q) = 1 and q = 3, read by rows.

%I A173045 #7 Feb 19 2021 18:34:37
%S A173045 1,1,1,1,10,1,1,29,29,1,1,84,6566,84,1,1,247,14348916,14348916,247,1,
%T A173045 1,734,282429536495,150094635296999140,282429536495,734,1,1,2193,
%U A173045 50031545098999727,2503155504993241601315571986085883,2503155504993241601315571986085883,50031545098999727,2193,1
%N A173045 Triangle T(n, k, q) = binomial(n, k) - 1 + q^(n*binomial(n-2, k-1)) with T(n, 0, q) = T(n, n, q) = 1 and q = 3, read by rows.
%H A173045 G. C. Greubel, <a href="/A173045/b173045.txt">Rows n = 0..11 of the triangle, flattened</a>
%F A173045 T(n, k, q) = binomial(n, k) - 1 + q^(n*binomial(n-2, k-1)) with T(n, 0, q) = T(n, n, q) = 1 and q = 3.
%F A173045 Sum_{k=0..n} T(n, k, 3) = A000295(n) + Sum_{k=0..n} 3^(n*binomial(n-2, k-1)). - _G. C. Greubel_, Feb 19 2021
%e A173045 Triangle begins as:
%e A173045   1;
%e A173045   1,   1;
%e A173045   1,  10,            1;
%e A173045   1,  29,           29,                  1;
%e A173045   1,  84,         6566,                 84,            1;
%e A173045   1, 247,     14348916,           14348916,          247,   1;
%e A173045   1, 734, 282429536495, 150094635296999140, 282429536495, 734, 1;
%t A173045 T[n_, k_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1 + q^(n*Binomial[n-2, k-1])];
%t A173045 Table[t[n, k, 3], {n,0,9}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 19 2021 *)
%o A173045 (Sage)
%o A173045 def T(n,k,q):
%o A173045     if (k==0 or k==n): return 1
%o A173045     else: return binomial(n,k) -1 +q^(n*binomial(n-2, k-1))
%o A173045 flatten([[T(n,k,3) for k in (0..n)] for n in (0..9)]) # _G. C. Greubel_, Feb 19 2021
%o A173045 (Magma)
%o A173045 T:= func< n,k,q | k eq 0 or k eq n select 1 else Binomial(n,k) -1 +q^(n*Binomial(n-2, k-1)) >;
%o A173045 [T(n,k,3): k in [0..n], n in [0..9]]; // _G. C. Greubel_, Feb 19 2021
%Y A173045 Cf. A132044 (q=0), A007318 (q=1), A173043 (q=2), this sequence (q=3).
%Y A173045 Cf. A000295.
%K A173045 nonn,tabl
%O A173045 0,5
%A A173045 _Roger L. Bagula_, Feb 08 2010
%E A173045 Edited by _G. C. Greubel_, Feb 19 2021