%I #30 Jul 11 2017 04:52:24
%S 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,
%T 34,333,312,196,93,33,8,1,55,1010,976,643,331,135,42,9,1,89,3202,3147,
%U 2137,1161,518,187,52,10,1,144,10504,10415,7213,4066,1929,768,250,63,11,1
%N Riordan array ( (1+x)/(1-x-x^2), x*A000108(x) ).
%H G. C. Greubel, <a href="/A185675/b185675.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H E. Deutsch, L. Ferrari and S. Rinaldi, <a href="http://arxiv.org/abs/math/0702638">Production Matrices and Riordan arrays</a>, arXiv:math/0702638 [math.CO], 2007.
%F 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.
%F R(n,0) = A000045(n+2).
%e Triangle begins:
%e 1;
%e 2, 1;
%e 3, 3, 1;
%e 5, 7, 4, 1;
%e 8, 17, 12, 5, 1;
%e 13, 43, 35, 18, 6, 1;
%e 21, 116, 103, 60, 25, 7, 1;
%e 34, 333, 312, 196, 93, 33, 8, 1;
%e Production matrix begins:
%e 2, 1;
%e -1, 1, 1;
%e 2, 1, 1, 1;
%e -3, 1, 1, 1, 1;
%e 5, 1, 1, 1, 1, 1;
%e -8, 1, 1, 1, 1, 1, 1;
%e 13, 1, 1, 1, 1, 1, 1, 1;
%e -21, 1, 1, 1, 1, 1, 1, 1, 1;
%e ... _Philippe Deléham_, Sep 21 2014
%p 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:
%p seq(seq(A185675(n,k),k=0..n),n=0..15) ; # _R. J. Mathar_, Feb 10 2011
%t 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 *)
%K nonn,tabl
%O 0,2
%A _Vladimir Kruchinin_, Feb 09 2011