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%I #16 Jan 07 2024 02:16:11
%S 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 101,46,11,12,57,147,231,231,147,57,12,13,69,204,378,462,378,204,69,
%U 13,14,82,273,582,840,840,582,273,82,14,15,96,355,855,1422,1680,1422,855,355,96,15
%N Elements in the 5-Pascal triangle A028313 that are not 1.
%H G. C. Greubel, <a href="/A028314/b028314.txt">Rows n = 0..50 of the triangle, flattened</a>
%F From _G. C. Greubel_, Jan 06 2024: (Start)
%F T(n, k) = binomial(n+2, k+1) + 3*binomial(n, k).
%F T(n, n-k) = T(n, k).
%F T(n, 0) = T(n, n) = A000027(n+5).
%F T(n, 1) = T(n, n-1) = A051936(n+4).
%F T(n, 2) = T(n, n-2) = A051937(n+3).T(2*n, n) = A028322(n+1).
%F Sum_{k=0..n} T(n, k) = A176448(n).
%F Sum_{k=0..n} (-1)^k * T(n, k) = 1 + (-1)^n + 3*[n=0].
%F Sum_{k=0..n} T(n-k, k) = A022112(n+1) - (3-(-1)^n)/2.
%F Sum_{k=0..n} (-1)^k * T(n-k, k) = 4*A010892(n) - 2*A121262(n+1) - (3 - (-1)^n)/2. (End)
%e Triangle begins as:
%e 5;
%e 6, 6;
%e 7, 12, 7;
%e 8, 19, 19, 8;
%e 9, 27, 38, 27, 9;
%e 10, 36, 65, 65, 36, 10;
%e 11, 46, 101, 130, 101, 46, 11;
%e 12, 57, 147, 231, 231, 147, 57, 12;
%e 13, 69, 204, 378, 462, 378, 204, 69, 13;
%t A028314[n_, k_]:= Binomial[n+2,k+1] + 3*Binomial[n,k];
%t Table[A028314[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 06 2024 *)
%o (Magma)
%o A028314:= func< n,k | Binomial(n+2,k+1) + 3*Binomial(n,k) >
%o [A028314(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 06 2024
%o (SageMath)
%o def A028314(n,k): return binomial(n+2,k+1) + 3*binomial(n,k)
%o flatten([[A028314(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jan 06 2024
%Y Cf. A000027, A010892, A022112, A028313, A028322,
%Y Cf. A051936, A051937, A121262, A176448.
%K nonn,tabl
%O 0,1
%A _Mohammad K. Azarian_
%E More terms from _James A. Sellers_
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