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1, -2, 1, 5, -5, 1, -13, 19, -8, 1, 34, -65, 42, -11, 1, -89, 210, -183, 74, -14, 1, 233, -654, 717, -394, 115, -17, 1, -610, 1985, -2622, 1825, -725, 165, -20, 1, 1597, -5911, 9134, -7703, 3885, -1203, 224, -23, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Left border (unsigned) = odd-indexed Fibonacci numbers. Left border (unsigned) of A123965 = even-indexed Fibonacci numbers.
Subtriangle of the triangle T(n,k) given by [0,-2,-1/2,-1/2,0,0,0,0,...] DELTA [1,0,1/2,-1/2,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 02 2007
Reversals = bisection of triangle A152063: (1; 1,2; 1,5,5; ...) having the following property: Product_{k=1..floor((n-1)/2)} (1 + 4*cos^2 k*2Pi/n) = the odd-indexed Fibonacci numbers. Example: x^3 - 8x^2 - 19x + 13 relates to the heptagon, and with k=1,2,3,..., the product = 13. - Gary W. Adamson, Aug 15 2010
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LINKS
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FORMULA
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T(n,k) = (-1)^(n+k)*Sum_{m=k..n} binomial(m,k)*binomial(m+n,2*m). - Wadim Zudilin, Jan 11 2012
G.f.: (1+x)*x*y/(1+3*x+x^2-x*y). - R. J. Mathar, Aug 11 2015
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EXAMPLE
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First few rows of the triangle are:
1;
-2, 1;
5, -5, 1;
-13, 19, -8, 1;
34, -65, 42, -11, 1;
-89, 210, -183, 74, -14, 1;
...
Triangle (n >= 0 and 0 <= k <= n) [0,-2,-1/2,-1/2,0,0,0,0,0,...] DELTA [1,0,1/2,-1/2,0,0,0,0,0,...] begins:
1;
0, 1;
0, -2, 1;
0, 5, -5, 1;
0, -13, 19, -8, 1;
0, 34, -65, 42, -11, 1;
0, -89, 210, -183, 74, -14, 1;
0, 233, -654, 717, -394, 115, -17, 1;
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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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