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A028331 Elements to the right of the central elements of the even-Pascal triangle A028326 that are not 2.

%I A028331 #13 Jul 02 2025 16:01:56
%S A028331 6,8,20,10,30,12,70,42,14,112,56,16,252,168,72,18,420,240,90,20,924,
%T A028331 660,330,110,22,1584,990,440,132,24,3432,2574,1430,572,156,26,6006,
%U A028331 4004,2002,728,182,28,12870,10010,6006,2730,910,210,30,22880,16016
%N A028331 Elements to the right of the central elements of the even-Pascal triangle A028326 that are not 2.
%H A028331 G. C. Greubel, <a href="/A028331/b028331.txt">Rows n = 0..100 of the irregular triangle, flattened</a>
%F A028331 From _G. C. Greubel_, Jul 14 2024: (Start)
%F A028331 T(n, k) = 2*binomial(n+3, k+2 + floor((n+1)/2)).
%F A028331 Sum_{k=0..floor(n/2)} T(n, k) = A272514(n+3).
%F A028331 Sum_{k=0..n} (-1)^k*T(2*n, k) = 2*A286033(n+2).
%F A028331 Sum_{k=0..n} (-1)^k*T(2*n+1, k) = binomial(2*n+4, n+2) + 2*(-1)^n.
%F A028331 (End)
%e A028331 This sequence represents the following portion of A028330(n,k), with x being the elements of A028329(n):
%e A028331   x;
%e A028331   .,  .;
%e A028331   .,  x,  .;
%e A028331   .,  .,  6,  .;
%e A028331   .,  .,  x,  8,  .;
%e A028331   .,  .,  ., 20, 10,   .;
%e A028331   .,  .,  .,  x, 30,  12,   .;
%e A028331   .,  .,  .,  ., 70,  42,  14,    .;
%e A028331   .,  .,  .,  .,  x, 112,  56,   16,   .;
%e A028331   .,  .,  .,  .,  ., 252, 168,   72,  18,   .;
%e A028331   .,  .,  .,  .,  .,   x, 420,  240,  90,  20,   .;
%e A028331   .,  .,  .,  .,  .,   ., 924,  660, 330, 110,  22,  .;
%e A028331   .,  .,  .,  .,  .,   .,   x, 1584, 990, 440, 132, 24, .;
%e A028331 As an irregular triangle:
%e A028331     6;
%e A028331     8;
%e A028331    20,  10;
%e A028331    30,  12;
%e A028331    70,  42,  14;
%e A028331   112,  56,  16;
%e A028331   252, 168,  72,  18;
%e A028331   420, 240,  90,  20;
%e A028331   924, 660, 330, 110, 22;
%t A028331 Table[2*Binomial[n+3, k+2 +Floor[(n+1)/2]], {n,0,12}, {k,0,Floor[n/2] }]//Flatten (* _G. C. Greubel_, Jul 14 2024 *)
%o A028331 (Magma)
%o A028331 [2*Binomial(n+3,k): k in [Floor((n+5)/2)..n+2], n in [0..12]]; // _G. C. Greubel_, Jul 14 2024
%o A028331 (SageMath)
%o A028331 def A028326(n,k): return 2*binomial(n, k)
%o A028331 flatten([[A028326(n+1,k) for k in range(((n+3)//2), n+1)] for n in range(21)]) # _G. C. Greubel_, Jul 14 2024
%Y A028331 Cf. A028326, A028327, A028328, A028329, A028330, A028332.
%Y A028331 Cf. A272514, A286033.
%K A028331 nonn,tabf
%O A028331 0,1
%A A028331 _Mohammad K. Azarian_
%E A028331 More terms from _James Sellers_