A156580 - OEIS

%I #6 Jan 05 2022 10:36:40

%S 1,1,1,1,2,1,1,4,4,1,1,6,7,6,1,1,8,10,10,8,1,1,10,12,13,12,10,1,1,12,

%T 14,16,16,14,12,1,1,14,16,18,19,18,16,14,1,1,16,18,20,22,22,20,18,16,

%U 1,1,18,20,22,24,25,24,22,20,18,1,1,20,22,24,26,28,28,26,24,22,20,1

%N Triangle T(n, k) = binomial(n-1, k-1) for n < 4, 2*(n + k - 4) + (-1)^n if k = (n+1)/2, 2*(n + k - 4) if k < floor(n/2) + 1, otherwise 2*(2*n - k - 3), with T(n, 0) = T(n, n) = 1, read by rows.

%H G. C. Greubel, <a href="/A156580/b156580.txt">Rows n = 1..50 of the triangle, flattened</a>

%F T(n, k) = binomial(n-1, k-1) for n < 4, 2*(n + k - 4) + (-1)^n if k = (n+1)/2, 2*(n + k - 4) if k < floor(n/2) + 1, otherwise 2*(2*n - k - 3), with T(n, 0) = T(n, n) = 1.

%F T(2*n, k) = T(2*n, 2*n-k+1).

%e Triangle begins as:

%e   1;

%e   1,  1;

%e   1,  2,  1;

%e   1,  4,  4,  1;

%e   1,  6,  7,  6,  1;

%e   1,  8, 10, 10,  8,  1;

%e   1, 10, 12, 13, 12, 10,  1;

%e   1, 12, 14, 16, 16, 14, 12,  1;

%e   1, 14, 16, 18, 19, 18, 16, 14,  1;

%e   1, 16, 18, 20, 22, 22, 20, 18, 16,  1;

%e   1, 18, 20, 22, 24, 25, 24, 22, 20, 18,  1;

%e   1, 20, 22, 24, 26, 28, 28, 26, 24, 22, 20, 1;

%t T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, If[n<4, Binomial[n-1, k-1], If[k==(n+1)/2, 2*(n+k-4) + (-1)^n, If[Floor[n/2]>(k-1), 2*(n+k-4), 2*(2*n-k-3) ]]]];

%t Table[T[n, k], {n,15}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Jan 04 2022 *)

%o (Sage)

%o def T(n,k):

%o     if (k==1 or k==n): return 1

%o     elif (n<4): return binomial(n-1, k-1)

%o     elif (k==(n+1)/2): return 2*(n+k-4) + (-1)^n

%o     elif (k<(n//2)+1): return 2*(n+k-4)

%o     else: return 2*(2*n - k - 3)

%o [[T(n,k) for k in (1..n)] for n in (1..15)] # _G. C. Greubel_, Jan 04 2022

%K nonn,tabl

%O 1,5

%A _Roger L. Bagula_, Feb 10 2009

%E Edited by _G. C. Greubel_, Jan 04 2022