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A172176
Triangle T(n, k) = 1 + (n + k - n*k)*(2*n - k - n*(n-k)), read by rows.
1
1, 2, 2, 1, 2, 1, -8, 0, 0, -8, -31, -4, 5, -4, -31, -74, -10, 22, 22, -10, -74, -143, -18, 57, 82, 57, -18, -143, -244, -28, 116, 188, 188, 116, -28, -244, -383, -40, 205, 352, 401, 352, 205, -40, -383, -566, -54, 330, 586, 714, 714, 586, 330, -54, -566
OFFSET
0,2
FORMULA
T(n, k) = 1 + (n-(n-1)*k)*(n-(n-1)*(n-k)).
T(n, n-k) = T(n, k).
T(n, 0) = 1 - A027620(n-3).
T(n, 1) = -A028552(n-3).
T(n, 2) = A033445(n-2).
Sum_{k=0..n} T(n, k) = (n+1)*(n^4 - 9*n^3 + 15*n^2 - n + 6)/6.
EXAMPLE
Triangle begins as:
1;
2, 2;
1, 2, 1;
-8, 0, 0, -8;
-31, -4, 5, -4, -31;
-74, -10, 22, 22, -10, -74;
-143, -18, 57, 82, 57, -18, -143;
-244, -28, 116, 188, 188, 116, -28, -244;
-383, -40, 205, 352, 401, 352, 205, -40, -383;
-566, -54, 330, 586, 714, 714, 586, 330, -54, -566;
-799, -70, 497, 902, 1145, 1226, 1145, 902, 497, -70, -799;
MAPLE
A172176:= proc(n, m) 1+(n+m-n*m)*(2*n-m-n*(n-m)); end proc:
seq(seq(A172176(n, m), m=0..n), n=0..12);
MATHEMATICA
T[n_, k_]= 1 + (n-(n-1)*k)*(n-(n-1)*(n-k));
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
PROG
(Magma) [1 + (n-(n-1)*k)*(n-(n-1)*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 26 2022
(SageMath)
def A172176(n, k): return 1 + (n-(n-1)*k)*(n-(n-1)*(n-k))
flatten([[A172176(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 26 2022
CROSSREFS
KEYWORD
sign,tabl,easy
AUTHOR
Roger L. Bagula, Jan 28 2010
STATUS
approved