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A168623 Triangle read by rows: T(n, k) = [x^k]( 9*(1+x)^n - 8*(1 + x^n) ), with T(0, 0) = 1.

%I A168623 #9 Apr 10 2025 01:35:59
%S A168623 1,1,1,1,18,1,1,27,27,1,1,36,54,36,1,1,45,90,90,45,1,1,54,135,180,135,
%T A168623 54,1,1,63,189,315,315,189,63,1,1,72,252,504,630,504,252,72,1,1,81,
%U A168623 324,756,1134,1134,756,324,81,1,1,90,405,1080,1890,2268,1890,1080,405,90,1
%N A168623 Triangle read by rows: T(n, k) = [x^k]( 9*(1+x)^n - 8*(1 + x^n) ), with T(0, 0) = 1.
%H A168623 G. C. Greubel, <a href="/A168623/b168623.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A168623 From _G. C. Greubel_, Apr 09 2025: (Start)
%F A168623 T(n, k) = binomial(n,k)*( [k=0] + 9*[0 < k < n] + [k=n] ).
%F A168623 T(n, n-k) = T(n, k).
%F A168623 Sum_{k=0..n} T(n, k) = 9*2^n - 16 + 8*[n=0].
%F A168623 Sum_{k=0..n} (-1)^k*T(n, k) = -8*(1 + (-1)^n) + 17*[n=0].
%F A168623 Sum_{k=0..floor(n/2)} T(n-k, k) = 9*A000045(n+1) - 4*(3 + (-1)^n) + 8*[n=0]. (End)
%e A168623 Triangle begins as:
%e A168623   1;
%e A168623   1,  1;
%e A168623   1, 18,   1;
%e A168623   1, 27,  27,    1;
%e A168623   1, 36,  54,   36,    1;
%e A168623   1, 45,  90,   90,   45,    1;
%e A168623   1, 54, 135,  180,  135,   54,    1;
%e A168623   1, 63, 189,  315,  315,  189,   63,    1;
%e A168623   1, 72, 252,  504,  630,  504,  252,   72,   1;
%e A168623   1, 81, 324,  756, 1134, 1134,  756,  324,  81,  1;
%e A168623   1, 90, 405, 1080, 1890, 2268, 1890, 1080, 405, 90, 1;
%t A168623 (* First program *)
%t A168623 p[x_, n_]:= With[{m=4}, If[n==0, 1, (2*m+1)(1+x)^n - 2*m*(1+x^n)]];
%t A168623 Table[CoefficientList[p[x,n], x], {n,0,10}]//Flatten
%t A168623 (* Second program *)
%t A168623 A168623[n_, k_]:= If[k==0 || k==n, 1, 9*Binomial[n,k]];
%t A168623 Table[A168623[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 09 2025 *)
%o A168623 (Magma)
%o A168623 A168623:= func< n,k | k eq 0 or k eq n select 1 else 9*Binomial(n,k) >;
%o A168623 [A168623(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 09 2025
%o A168623 (SageMath)
%o A168623 def A168623(n,k):
%o A168623     if (k==0 or k==n): return 1
%o A168623     else: return 9*binomial(n,k)
%o A168623 print(flatten([[A168623(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Apr 09 2025
%Y A168623 Cf. A000045, A132047.
%K A168623 nonn,tabl
%O A168623 0,5
%A A168623 _Roger L. Bagula_ and _Gary W. Adamson_, Dec 01 2009
%E A168623 Keyword:tabl and row sum formula added - The Assoc. Editors of the OEIS, Dec 05 2009