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A144431
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Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n,1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = -1.
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21
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1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, 2, -2, 1, 1, -3, 2, 2, -3, 1, 1, -4, 7, -8, 7, -4, 1, 1, -5, 9, -5, -5, 9, -5, 1, 1, -6, 16, -26, 30, -26, 16, -6, 1, 1, -7, 20, -28, 14, 14, -28, 20, -7, 1, 1, -8, 29, -64, 98, -112, 98, -64, 29, -8, 1, 1, -9, 35, -75, 90, -42, -42, 90, -75, 35, -9, 1
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OFFSET
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1,12
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COMMENTS
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Row sums are: {1, 2, 2, 0, 0, 0, 0, 0, 0, 0, ...}.
For m = ...,-1,0,1,2 we get ..., A144431, A007318 (Pascal), A008292, A060187, ..., so this might be called a sub-Pascal triangle.
The triangle starts off like A098593, but is different further on.
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LINKS
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FORMULA
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T(n,k) = (m*n - m*k + 1)*T(n-1, k-1) + (m*k - (m-1))*T(n-1, k) with T(n, 1) = T(n, n) = 1 and m = -1.
T(n, n-k) = T(n, k).
T(n, k) = (-1)^(k-1)*binomial(n-3, k-1) + (-1)^(n+k)*binomial(n-3, k-3) with T(1, k) = T(2, k) = 1.
Sum_{k=1..n} T(n, k) = [n==1] + 2*[n==2] + 2*[n==3] + (1-(-1)^n)*0^(n-3)*[n>3]. (End)
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EXAMPLE
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Triangle begins:
1;
1, 1;
1, 0, 1;
1, -1, -1, 1;
1, -2, 2, -2, 1;
1, -3, 2, 2, -3, 1;
1, -4, 7, -8, 7, -4, 1;
1, -5, 9, -5, -5, 9, -5, 1;
1, -6, 16, -26, 30, -26, 16, -6, 1;
1, -7, 20, -28, 14, 14, -28, 20, -7, 1;
...
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MAPLE
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T:=proc(n, k, l) option remember;
if (n=1 or k=1 or k=n) then 1 else
(l*n-l*k+1)*T(n-1, k-1, l)+(l*k-l+1)*T(n-1, k, l); fi; end;
for n from 1 to 15 do lprint([seq(T(n, k, -1), k=1..n)]); od; # N. J. A. Sloane, May 08 2013
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MATHEMATICA
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m=-1;
T[n_, 1]:= 1; T[n_, n_]:= 1;
T[n_, k_]:= (m*n-m*k+1)*T[n-1, k-1] + (m*k - (m - 1))*T[n-1, k];
Table[T[n, k], {n, 15}, {k, n}]//Flatten
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PROG
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(Sage)
if (n<3): return 1
else: return (-1)^(k-1)*binomial(n-3, k-1) + (-1)^(n+k)*binomial(n-3, k-3)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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