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A185675 Riordan array ( (1+x)/(1-x-x^2), x*A000108(x) ).

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