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A225372
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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 = -2.
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6
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1, 1, 1, 1, -2, 1, 1, -1, -1, 1, 1, -4, 6, -4, 1, 1, -3, 2, 2, -3, 1, 1, -6, 15, -20, 15, -6, 1, 1, -5, 9, -5, -5, 9, -5, 1, 1, -8, 28, -56, 70, -56, 28, -8, 1, 1, -7, 20, -28, 14, 14, -28, 20, -7, 1, 1, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1
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OFFSET
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1,5
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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 = -2.
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EXAMPLE
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Triangle begins:
1;
1, 1;
1, -2, 1;
1, -1, -1, 1;
1, -4, 6, -4, 1;
1, -3, 2, 2, -3, 1;
1, -6, 15, -20, 15, -6, 1;
1, -5, 9, -5, -5, 9, -5, 1;
1, -8, 28, -56, 70, -56, 28, -8, 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 14 do lprint([seq(T(n, k, -2), k=1..n)]); od;
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MATHEMATICA
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T[n_, k_, l_] := T[n, k, l] = If[n == 1 || k == 1 || k == n, 1, (l*n-l*k+1)*T[n-1, k-1, l]+(l*k-l+1)*T[n-1, k, l]]; Table[T[n, k, -2], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 09 2014, translated from Maple *)
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PROG
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(Magma)
function T(n, k, m)
if k eq 1 or k eq n then return 1;
else return (m*(n-k)+1)*T(n-1, k-1, m) + (m*k-m+1)*T(n-1, k, m);
end if; return T;
end function;
A225372:= func< n, k | T(n, k, -2) >;
(Sage)
@CachedFunction
def T(n, k, m):
if (k==1 or k==n): return 1
else: return (m*(n-k)+1)*T(n-1, k-1, m) + (m*k-m+1)*T(n-1, k, m)
def A225372(n, k): return T(n, k, -2)
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CROSSREFS
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For m = ...,-2,-1,0,1,2,3,4,5,6,7,8, ... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142560, A142561, A142562, A167884, ...
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KEYWORD
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AUTHOR
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STATUS
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approved
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