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A139375
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A Fibonacci-Catalan triangle. Also called the Fibonacci triangle.
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6
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1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 5, 12, 9, 4, 1, 8, 31, 26, 14, 5, 1, 13, 85, 77, 46, 20, 6, 1, 21, 248, 235, 150, 73, 27, 7, 1, 34, 762, 741, 493, 258, 108, 35, 8, 1, 55, 2440, 2406, 1644, 903, 410, 152, 44, 9, 1, 89, 8064, 8009
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| First column is the Fibonacci numbers A000045(n+1). The second column is A090826.
Row sums are A090826(n+1). Diagonal sums are A139376. Inverse array is (1-x+2x^3-x^4,x(1-x)), A201167.
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REFERENCES
| He, Tian-Xiao, and Sprugnoli, Renzo; Sequence characterization of Riordan arrays. Discrete Math. 309 (2009), no. 12, 3962-3974. [From N. J. A. Sloane, Nov 26, 2011]
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FORMULA
| Riordan array (1/(1-x-x^2), xc(x)), c(x) the g.f. of A000108.
T(n,k)=k*sum(i=0..n-k,fibonacci(i+1)*binomial(2*(n-i)-k-1,n-i-1)/(n-i)) if k>0, and fibonacci(n+1) if k=0. [From Vladimir Kruchinin, Mar 09 2011]
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EXAMPLE
| Triangle begins
1,
1, 1,
2, 2, 1,
3, 5, 3, 1,
5, 12, 9, 4, 1,
8, 31, 26, 14, 5, 1,
13, 85, 77, 46, 20, 6, 1,
21, 248, 235, 150, 73, 27, 7, 1,
34, 762, 741, 493, 258, 108, 35, 8, 1
The production matrix for this array is
1, 1,
1, 1, 1,
-1, 1, 1, 1,
0, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1
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CROSSREFS
| Sequence in context: A106196 A037027 A182810 * A106198 A054336 A079956
Adjacent sequences: A139372 A139373 A139374 * A139376 A139377 A139378
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Apr 15 2008
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EXTENSIONS
| Alternative name added by N. J. A. Sloane, Nov 27 2011
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