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A156644
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Mirror image of triangle A080233.
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6
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1, 0, 1, -1, 1, 1, -2, 0, 2, 1, -3, -2, 2, 3, 1, -4, -5, 0, 5, 4, 1, -5, -9, -5, 5, 9, 5, 1, -6, -14, -14, 0, 14, 14, 6, 1, -7, -20, -28, -14, 14, 28, 20, 7, 1, -8, -27, -48, -42, 0, 42, 48, 27, 8, 1, -9, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,7
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COMMENTS
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LINKS
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FORMULA
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T(n,k) = ((2*k-n+1)/(k+1))*binomial(n,k).
T(n,k) = T(n-1,k-1) + T(n-1,k), k>0, with T(n,0) = 1-n = A024000(n), T(n,n) = 1.
T(n,k) = binomial(n,k) - binomial(n,k+1) = Sum_{i=-k-1..k+1} (-1)^(i+1) * binomial(n,k+1+i) * binomial(n+2,k+1-i). - Mircea Merca, Apr 28 2012
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EXAMPLE
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Triangle begins as:
1;
0, 1;
-1, 1, 1;
-2, 0, 2, 1;
-3, -2, 2, 3, 1;
-4, -5, 0, 5, 4, 1; ...
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MATHEMATICA
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Table[Binomial[n, k] -Binomial[n, k+1], {n, 0, 10}, {k, 0, n}]//Flatten (* Michael De Vlieger, Nov 24 2016 *)
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PROG
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(Sage)
def A156644(n, k): return ((2*k-n+1)/(k+1))*binomial(n, k)
(Magma)
A156644:= func< n, k | ((2*k-n+1)/(k+1))*Binomial(n, k) >;
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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