OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices and Riordan arrays, arXiv:math/0702638 [math.CO], 2007.
FORMULA
R(n,k) = k*Sum_{i=0..n-k}(Sum_{j=1..i+1}binomial(j,i+1-j))*binomial(2*(n-i)-k-1,n-i-1)/(n-i), k>0.
R(n,0) = A000045(n+2).
EXAMPLE
Triangle begins:
1;
2, 1;
3, 3, 1;
5, 7, 4, 1;
8, 17, 12, 5, 1;
13, 43, 35, 18, 6, 1;
21, 116, 103, 60, 25, 7, 1;
34, 333, 312, 196, 93, 33, 8, 1;
Production matrix begins:
2, 1;
-1, 1, 1;
2, 1, 1, 1;
-3, 1, 1, 1, 1;
5, 1, 1, 1, 1, 1;
-8, 1, 1, 1, 1, 1, 1;
13, 1, 1, 1, 1, 1, 1, 1;
-21, 1, 1, 1, 1, 1, 1, 1, 1;
... Philippe Deléham, Sep 21 2014
MAPLE
A185675 := proc(n, k) if n = k then 1; elif k = 0 then combinat[fibonacci](n+2) ; else k*add(1/(n-i)*add(binomial(j, i+1-j)*binomial(2*n-2*i-k-1, n-i-1), j=1..i+1), i=0..n-k) ; end if; end proc:
seq(seq(A185675(n, k), k=0..n), n=0..15) ; # R. J. Mathar, Feb 10 2011
MATHEMATICA
r[n_, k_] := k*Sum[Binomial[2*(n - i) - k - 1, n - i - 1]*Fibonacci[i + 2]/(n - i), {i, 0, n - k}]; r[n_, 0] := Fibonacci[n + 2]; r[n_, n_] := 1; Table[r[n, k], {n, 0, 3}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 21 2013; modified by G. C. Greubel, Jul 10 2017 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, Feb 09 2011
STATUS
approved