cp's OEIS Frontend

A173077 Triangle T(n, k, q) = binomial(n, k) - 1 + q^floor(n/2)*binomial(n-2, k-1) with T(n, 0, q) = T(n, n, q) = 1 and q = 3, read by rows.

%I A173077 #14 Sep 08 2022 08:45:50
%S A173077 1,1,1,1,4,1,1,5,5,1,1,12,23,12,1,1,13,36,36,13,1,1,32,122,181,122,32,
%T A173077 1,1,33,155,304,304,155,33,1,1,88,513,1270,1689,1270,513,88,1,1,89,
%U A173077 602,1784,2960,2960,1784,602,89,1,1,252,1988,6923,13817,17261,13817,6923,1988,252,1
%N A173077 Triangle T(n, k, q) = binomial(n, k) - 1 + q^floor(n/2)*binomial(n-2, k-1) with T(n, 0, q) = T(n, n, q) = 1 and q = 3, read by rows.
%H A173077 G. C. Greubel, <a href="/A173077/b173077.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A173077 T(n, k, q) = binomial(n, k) - 1 + q^floor(n/2)*binomial(n-2, k-1) with T(n, 0, q) = T(n, n, q) = 1 and q = 3.
%e A173077 Triangle starts:
%e A173077   1;
%e A173077   1,   1;
%e A173077   1,   4,    1;
%e A173077   1,   5,    5,    1;
%e A173077   1,  12,   23,   12,     1;
%e A173077   1,  13,   36,   36,    13,     1;
%e A173077   1,  32,  122,  181,   122,    32,     1;
%e A173077   1,  33,  155,  304,   304,   155,    33,    1;
%e A173077   1,  88,  513, 1270,  1689,  1270,   513,   88,    1;
%e A173077   1,  89,  602, 1784,  2960,  2960,  1784,  602,   89,   1;
%e A173077   1, 252, 1988, 6923, 13817, 17261, 13817, 6923, 1988, 252, 1;
%e A173077   ...
%e A173077 Row sums: 1, 2, 6, 12, 49, 100, 491, 986, 5433, 10872, 63223, ...
%t A173077 T[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1 + 3^Floor[n/2] Binomial[n-2, k- 1]];
%t A173077 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
%o A173077 (Magma)
%o A173077 T:= func< n,k,q | k eq 0 or k eq n select 1 else Binomial(n,k) + q^(Floor(n/2))*Binomial(n-2,k-1) -1 >;
%o A173077 [T(n,k,3): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 09 2021
%o A173077 (Sage)
%o A173077 def T(n,k,q): return 1 if (k==0 or k==n) else binomial(n,k) + q^(n//2)*binomial(n-2,k-1) -1
%o A173077 flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 09 2021
%Y A173077 Cf. A132044 (q=0), A173075 (q=1), A173076 (q=2), this sequence (q=3).
%K A173077 nonn,tabl
%O A173077 0,5
%A A173077 _Roger L. Bagula_, Feb 09 2010