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A028314 Elements in the 5-Pascal triangle A028313 that are not 1.

%I A028314 #25 Jul 02 2025 16:01:56
%S A028314 5,6,6,7,12,7,8,19,19,8,9,27,38,27,9,10,36,65,65,36,10,11,46,101,130,
%T A028314 101,46,11,12,57,147,231,231,147,57,12,13,69,204,378,462,378,204,69,
%U A028314 13,14,82,273,582,840,840,582,273,82,14,15,96,355,855,1422,1680,1422,855,355,96,15
%N A028314 Elements in the 5-Pascal triangle A028313 that are not 1.
%H A028314 G. C. Greubel, <a href="/A028314/b028314.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A028314 From _G. C. Greubel_, Jan 06 2024: (Start)
%F A028314 T(n, k) = binomial(n+2, k+1) + 3*binomial(n, k).
%F A028314 T(n, n-k) = T(n, k).
%F A028314 T(n, 0) = T(n, n) = A000027(n+5).
%F A028314 T(n, 1) = T(n, n-1) = A051936(n+4).
%F A028314 T(n, 2) = T(n, n-2) = A051937(n+3).T(2*n, n) = A028322(n+1).
%F A028314 Sum_{k=0..n} T(n, k) = A176448(n).
%F A028314 Sum_{k=0..n} (-1)^k * T(n, k) = 1 + (-1)^n + 3*[n=0].
%F A028314 Sum_{k=0..n} T(n-k, k) = A022112(n+1) - (3-(-1)^n)/2.
%F A028314 Sum_{k=0..n} (-1)^k * T(n-k, k) = 4*A010892(n) - 2*A121262(n+1) - (3 - (-1)^n)/2. (End)
%F A028314 G.f.: (5 - 4*x - 4*x*y + 3*x^2*y)/((1 - x)*(1 - x*y)*(1 - x - x*y)). - _Stefano Spezia_, Dec 06 2024
%e A028314 Triangle begins as:
%e A028314    5;
%e A028314    6,  6;
%e A028314    7, 12,   7;
%e A028314    8, 19,  19,   8;
%e A028314    9, 27,  38,  27,   9;
%e A028314   10, 36,  65,  65,  36,  10;
%e A028314   11, 46, 101, 130, 101,  46,  11;
%e A028314   12, 57, 147, 231, 231, 147,  57,  12;
%e A028314   13, 69, 204, 378, 462, 378, 204,  69,  13;
%t A028314 A028314[n_, k_]:= Binomial[n+2,k+1] + 3*Binomial[n,k];
%t A028314 Table[A028314[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 06 2024 *)
%o A028314 (Magma)
%o A028314 A028314:= func< n,k | Binomial(n+2,k+1) + 3*Binomial(n,k) >;
%o A028314 [A028314(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 06 2024
%o A028314 (SageMath)
%o A028314 def A028314(n,k): return binomial(n+2,k+1) + 3*binomial(n,k)
%o A028314 flatten([[A028314(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jan 06 2024
%Y A028314 Cf. A000027, A010892, A022112, A028313, A028322,
%Y A028314 Cf. A051936, A051937, A121262, A176448.
%K A028314 nonn,tabl
%O A028314 0,1
%A A028314 _Mohammad K. Azarian_
%E A028314 More terms from _James Sellers_