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A028323 Elements to the right of the central elements of the 5-Pascal triangle A028313.

%I A028323 #13 Jul 02 2025 16:01:56
%S A028323 1,1,6,1,7,1,19,8,1,27,9,1,65,36,10,1,101,46,11,1,231,147,57,12,1,378,
%T A028323 204,69,13,1,840,582,273,82,14,1,1422,855,355,96,15,1,3102,2277,1210,
%U A028323 451,111,16,1,5379,3487,1661,562,127,17,1,11583,8866,5148,2223,689,144,18,1
%N A028323 Elements to the right of the central elements of the 5-Pascal triangle A028313.
%H A028323 G. C. Greubel, <a href="/A028323/b028323.txt">Rows n = 0..100 of the irregular triangle, flattened</a>
%F A028323 From _G. C. Greubel_, Jan 05 2024: (Start)
%F A028323 a(n) = A028313(n, k), for 1 + floor(n/2) <= k <= n, n >= 0.
%F A028323 T(n, k) = binomial(n+1, k + floor((n+1)/2) + 1) + 3*binomial(n-1, k + floor((n+1)/2)) -3*[n=0], for 0 <= k <= floor(n/2), n >= 0. (End)
%e A028323 This sequence represents the following portion of A028313(n,k), with x being the elements of A028313(2*n,n):
%e A028323   x,
%e A028323   .,  1,
%e A028323   .,  x,  1,
%e A028323   .,  .,  6,   1,
%e A028323   .,  .,  x,   7,   1,
%e A028323   .,  ., ..,  19,   8,   1,
%e A028323   .,  ., ..,   x,  27,   9,   1,
%e A028323   ., .., ..,  ..,  65,  36,  10,  1,
%e A028323   ., .., .., ...,   x, 101,  46, 11,  1,
%e A028323   ., .., .., ..., ..., 231, 147, 57, 12, 1.
%e A028323 As an irregular triangle this sequence begins as:
%e A028323      1;
%e A028323      1;
%e A028323      6,    1;
%e A028323      7,    1;
%e A028323     19,    8,    1;
%e A028323     27,    9,    1;
%e A028323     65,   36,   10,   1;
%e A028323    101,   46,   11,   1;
%e A028323    231,  147,   57,  12,   1;
%e A028323    378,  204,   69,  13,   1;
%e A028323    840,  582,  273,  82,  14,  1;
%e A028323   1422,  855,  355,  96,  15,  1;
%e A028323   3102, 2277, 1210, 451, 111, 16,  1;
%t A028323 T[n_, k_]:= Binomial[n+1, k +Floor[(n+1)/2] +1] + 3*Binomial[n-1, k+ Floor[(n+1)/2]] -3*Boole[n==0];
%t A028323 Table[T[n,k], {n,0,16}, {k,0,Floor[n/2]}]//Flatten (* _G. C. Greubel_, Jan 05 2024 *)
%o A028323 (Magma)
%o A028323 A028323:= func< n,k | n eq 0 select 1 else Binomial(n+1, k + Floor((n+1)/2) + 1) + 3*Binomial(n-1, k + Floor((n+1)/2)) >;
%o A028323 [A028323(n,k): k in [0..Floor(n/2)], n in [0..16]]; // _G. C. Greubel_, Jan 05 2024
%o A028323 (SageMath)
%o A028323 def A028323(n,k): return binomial(n+1, k+1+(n+1)//2) + 3*binomial(n-1, k+((n+1)//2)) - 3*int(n==0)
%o A028323 flatten([[A028323(n,k) for k in range(1+(n//2))] for n in range(17)]) # _G. C. Greubel_, Jan 05 2024
%Y A028323 Cf. A028313.
%K A028323 nonn,easy,tabf
%O A028323 0,3
%A A028323 _Mohammad K. Azarian_
%E A028323 More terms from _James Sellers_