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A157654
Triangle T(n, k, m) = 1 if k = 0 or k = n, otherwise m*abs( (n-k)^(m-1) - k^(m-1) ), with m = 2, read by rows.
1
1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 4, 0, 4, 1, 1, 6, 2, 2, 6, 1, 1, 8, 4, 0, 4, 8, 1, 1, 10, 6, 2, 2, 6, 10, 1, 1, 12, 8, 4, 0, 4, 8, 12, 1, 1, 14, 10, 6, 2, 2, 6, 10, 14, 1, 1, 16, 12, 8, 4, 0, 4, 8, 12, 16, 1
OFFSET
0,8
COMMENTS
For the cases of m = 0, 1 the triangles reduce to T(n, k, m) = A103451(n, k). - G. C. Greubel, Dec 13 2021
FORMULA
T(n, k, m) = 1 if k = 0 or k = n, otherwise m*abs( (n-k)^(m-1) - k^(m-1) ), with m = 2.
From G. C. Greubel, Dec 13 2021: (Start)
Sum_{k=0..n} T(n, k, 2) = (-1)*[n==0] + A244800(n-1).
T(2*n, n, 2) = A000007(n). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 0, 1;
1, 2, 2, 1;
1, 4, 0, 4, 1;
1, 6, 2, 2, 6, 1;
1, 8, 4, 0, 4, 8, 1;
1, 10, 6, 2, 2, 6, 10, 1;
1, 12, 8, 4, 0, 4, 8, 12, 1;
1, 14, 10, 6, 2, 2, 6, 10, 14, 1;
1, 16, 12, 8, 4, 0, 4, 8, 12, 16, 1;
MATHEMATICA
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, m*Abs[(n-k)^(m-1) - k^(m-1)]];
Table[T[n, k, 2], {n, 0, 15}, {k, 0, n}]//Flatten
PROG
(Magma)
T:= func< n, k, q | k eq 0 or k eq n select 1 else q*Abs( (n-k)^(q-1) - k^(q-1) ) >;
[T(n, k, 2): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 13 2021
(Sage)
def A157684(n, k, q): return 1 if (k==0 or k==n) else q*abs((n-k)^(q-1) - k^(q-1))
flatten([[A157684(n, k, 2) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Dec 13 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Mar 03 2009
EXTENSIONS
Edited by G. C. Greubel, Dec 13 2021
STATUS
approved