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

0,4

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

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]

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

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

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 *)

Crossrefs

Sequence in context: A106196 A037027 A182810 * A106198 A202847 A054336

Adjacent sequences: A139372 A139373 A139374 * A139376 A139377 A139378

Keyword

easy,nonn,tabl

Author

Paul Barry, Apr 15 2008

Extensions

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

Status

approved