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A173043 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 = 2, read by rows.

%I A173043 #7 Feb 19 2021 18:34:28
%S A173043 1,1,1,1,5,1,1,10,10,1,1,19,261,19,1,1,36,32777,32777,36,1,1,69,
%T A173043 16777230,68719476755,16777230,69,1,1,134,34359738388,
%U A173043 1180591620717411303458,1180591620717411303458,34359738388,134,1
%N A173043 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 = 2, read by rows.
%H A173043 G. C. Greubel, <a href="/A173043/b173043.txt">Rows n = 0..12 of the triangle, flattened</a>
%F A173043 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 = 2.
%F A173043 Sum_{k=0..n} T(n, k, 2) = A000295(n) + Sum_{k=0..n} 2^(n*binomial(n-2, k-1)). - _G. C. Greubel_, Feb 19 2021
%e A173043 Triangle begins as:
%e A173043   1;
%e A173043   1,  1;
%e A173043   1,  5,        1;
%e A173043   1, 10,       10,           1;
%e A173043   1, 19,      261,          19,        1;
%e A173043   1, 36,    32777,       32777,       36,  1;
%e A173043   1, 69, 16777230, 68719476755, 16777230, 69, 1;
%t A173043 T[n_, k_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1 + q^(n*Binomial[n-2, k-1])];
%t A173043 Table[t[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 19 2021 *)
%o A173043 (Sage)
%o A173043 def T(n,k,q):
%o A173043     if (k==0 or k==n): return 1
%o A173043     else: return binomial(n,k) -1 +q^(n*binomial(n-2, k-1))
%o A173043 flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 19 2021
%o A173043 (Magma)
%o A173043 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 A173043 [T(n,k,2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 19 2021
%Y A173043 Cf. A132044 (q=0), A007318 (q=1), this sequence (q=2), A173045 (q=3).
%Y A173043 Cf. A000295.
%K A173043 nonn,tabl
%O A173043 0,5
%A A173043 _Roger L. Bagula_, Feb 08 2010
%E A173043 Edited by _G. C. Greubel_, Feb 19 2021