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A Fibonacci-Catalan triangle. Also called the Fibonacci triangle.
8

%I #29 Jul 23 2017 18:33:55

%S 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,

%T 248,235,150,73,27,7,1,34,762,741,493,258,108,35,8,1,55,2440,2406,

%U 1644,903,410,152,44,9,1,89,8064,8009

%N A Fibonacci-Catalan triangle. Also called the Fibonacci triangle.

%C First column is the Fibonacci numbers A000045(n+1). The second column is A090826.

%C Row sums are A090826(n+1). Diagonal sums are A139376. Inverse array is (1 - x + 2x^3 - x^4, x(1-x)), A201167.

%C Essentially A185937 with trailing zeros removed. - _Ralf Stephan_, Jan 01 2014

%H G. C. Greubel, <a href="/A139375/b139375.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H Tian-Xiao He and Renzo Sprugnoli, <a href="http://dx.doi.org/10.1016/j.disc.2008.11.021">Sequence characterization of Riordan arrays</a>, Discrete Math. 309 (2009), no. 12, 3962-3974. [_N. J. A. Sloane_, Nov 26 2011]

%F Riordan array (1/(1-x-x^2), xc(x)), c(x) the g.f. of A000108.

%F 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

%e Triangle begins

%e 1,

%e 1, 1,

%e 2, 2, 1,

%e 3, 5, 3, 1,

%e 5, 12, 9, 4, 1,

%e 8, 31, 26, 14, 5, 1,

%e 13, 85, 77, 46, 20, 6, 1,

%e 21, 248, 235, 150, 73, 27, 7, 1,

%e 34, 762, 741, 493, 258, 108, 35, 8, 1

%e The production matrix for this array is

%e 1, 1,

%e 1, 1, 1,

%e -1, 1, 1, 1,

%e 0, 1, 1, 1, 1,

%e 0, 1, 1, 1, 1, 1,

%e 0, 1, 1, 1, 1, 1, 1,

%e 0, 1, 1, 1, 1, 1, 1

%p RIORDAN := proc(d,h,n,k)

%p d*h^k ;

%p expand(%) ;

%p coeftayl(%,x=0,n) ;

%p end proc:

%p A139375 := proc(n,k)

%p RIORDAN(1/(1-x-x^2),(1-sqrt(1-4*x))/2,n,k) ;

%p end proc: # _R. J. Mathar_, Jul 09 2013

%t 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 *)

%K easy,nonn,tabl

%O 0,4

%A _Paul Barry_, Apr 15 2008

%E Alternative name added by _N. J. A. Sloane_, Nov 27 2011