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A168622 Triangle read by rows: T(n, k) = [x^k]( 7*(1+x)^n - 6*(1+x^n) ) with T(0, 0) = 1.

%I A168622 #12 Apr 10 2025 06:52:33
%S A168622 1,1,1,1,14,1,1,21,21,1,1,28,42,28,1,1,35,70,70,35,1,1,42,105,140,105,
%T A168622 42,1,1,49,147,245,245,147,49,1,1,56,196,392,490,392,196,56,1,1,63,
%U A168622 252,588,882,882,588,252,63,1,1,70,315,840,1470,1764,1470,840,315,70,1
%N A168622 Triangle read by rows: T(n, k) = [x^k]( 7*(1+x)^n - 6*(1+x^n) ) with T(0, 0) = 1.
%H A168622 G. C. Greubel, <a href="/A168622/b168622.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A168622 From _G. C. Greubel_, Apr 10 2025: (Start)
%F A168622 T(n, k) = 7*binomial(n, k), with T(n, 0) = T(n, n) = 1.
%F A168622 T(n, n-k) = T(n, k).
%F A168622 Sum_{k=0..n} T(n, k) = 2*A048489(n-1) + 6*[n=0].
%F A168622 Sum_{k=0..n} (-1)^k*T(n, k) = -6*(1 + (-1)^n) + 13*[n=0].
%F A168622 Sum_{k=0..floor(n/2)} T(n-k, k) = A022090(n+1) - 3*(3 + (-1)^n) + 6*[n=0].
%F A168622 Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (14/sqrt(3))*(-1)^n*cos((4*n+1)*Pi/6) - 6*(1 + (-1)^n*cos(n*Pi/2)) + 6*[n=0]. (End)
%e A168622 Triangle begins as:
%e A168622   1;
%e A168622   1,  1;
%e A168622   1, 14,   1;
%e A168622   1, 21,  21,   1;
%e A168622   1, 28,  42,  28,    1;
%e A168622   1, 35,  70,  70,   35,    1;
%e A168622   1, 42, 105, 140,  105,   42,    1;
%e A168622   1, 49, 147, 245,  245,  147,   49,   1;
%e A168622   1, 56, 196, 392,  490,  392,  196,  56,   1;
%e A168622   1, 63, 252, 588,  882,  882,  588, 252,  63,  1;
%e A168622   1, 70, 315, 840, 1470, 1764, 1470, 840, 315, 70, 1;
%t A168622 (* First program *)
%t A168622 p[x_, n_]:= With[{m=3}, If[n==0, 1, (2*m+1)(1+x)^n - 2*m*(1+x^n)]];
%t A168622 Table[CoefficientList[p[x,n], x], {n,0,12}]//Flatten
%t A168622 (* Second program *)
%t A168622 A168622[n_, k_]:= If[k==0 || k==n, 1, 7*Binomial[n,k]];
%t A168622 Table[A168622[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 10 2025 *)
%o A168622 (Magma)
%o A168622 A168622:= func< n,k | k eq 0 or k eq n select 1 else 7*Binomial(n,k) >;
%o A168622 [A168622(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 10 2025
%o A168622 (SageMath)
%o A168622 def A168622(n,k):
%o A168622     if k==0 or k==n: return 1
%o A168622     else: return 7*binomial(n,k)
%o A168622 print(flatten([[A168622(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Apr 10 2025
%Y A168622 Cf. A022090, A131115, A132047, A168620, A168623.
%Y A168622 Columns (essentially): A008589 (k=1), A024966 (k=2).
%K A168622 nonn,tabl,easy
%O A168622 0,5
%A A168622 _Roger L. Bagula_ and _Gary W. Adamson_, Dec 01 2009