A139375 A Fibonacci-Catalan triangle. Also called the Fibonacci triangle.

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

Offset: 0

Author

Paul Barry, Apr 15 2008

Keywords

• easy
• nonn
• tabl

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.

Essentially A185937 with trailing zeros removed. - Ralf Stephan, Jan 01 2014

Examples

			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
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• Tian-Xiao He and Renzo Sprugnoli, Sequence characterization of Riordan arrays, Discrete Math. 309 (2009), no. 12, 3962-3974. [_N. J. A. Sloane_, Nov 26 2011]

Programs

Maple

RIORDAN := proc(d,h,n,k)
    d*h^k ;
    expand(%) ;
    coeftayl(%,x=0,n) ;
end proc:
A139375 := proc(n,k)
    RIORDAN(1/(1-x-x^2),(1-sqrt(1-4*x))/2,n,k) ;
end proc: # R. J. Mathar, Jul 09 2013

Mathematica

T[n_, 0]:= Fibonacci[n + 1]; T[n_, k_]:= k*Sum[Fibonacci[i + 1]*Binomial[2*(n - i) - k - 1, n - i - 1]/(n - i), {i, 0, n - k}]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 20 2016 *)

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. - Vladimir Kruchinin, Mar 09 2011

Extensions

Alternative name added by N. J. A. Sloane, Nov 27 2011