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A185812 Riordan array ( 1/(1-x), x*A005043(x) ).

%I A185812 #21 Jul 15 2017 02:06:27
%S A185812 1,1,1,1,1,1,1,2,1,1,1,3,3,1,1,1,6,5,4,1,1,1,12,12,7,5,1,1,1,27,26,19,
%T A185812 9,6,1,1,1,63,63,43,27,11,7,1,1,1,154,153,110,63,36,13,8,1,1,1,386,
%U A185812 386,275,169,86,46,15,9,1,1
%N A185812 Riordan array ( 1/(1-x), x*A005043(x) ).
%H A185812 G. C. Greubel, <a href="/A185812/b185812.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185812 R(n,k) = k*Sum_{i=0..(n-k)} (Sum_{j=k..(n-i)} binomial(2*j-k-1,j-1) *(-1)^(n-j-i) *binomial(n-i,j))/(n-i), k>0.
%F A185812 R(n,0)=1.
%e A185812 Array begins:
%e A185812   1;
%e A185812   1,  1;
%e A185812   1,  1,  1;
%e A185812   1,  2,  1,  1;
%e A185812   1,  3,  3,  1,  1;
%e A185812   1,  6,  5,  4,  1,  1;
%e A185812   1, 12, 12,  7,  5,  1,  1;
%e A185812   1, 27, 26, 19,  9,  6,  1,  1;
%p A185812 A185812 := proc(n,k) if n = k  or k =0 then 1; else k*add(1/(n-i)*add(binomial(2*j-k-1,j-1) *(-1)^(n-j-i) *binomial(n-i,j),j=k..n-i),i=0..n-k) ; end if; end proc:
%p A185812 seq(seq(A185812(n,k),k=0..n),n=0..15) ; # _R. J. Mathar_, Feb 10 2011
%t A185812 r[n_, k_] := k*Sum[Binomial[2*j - k - 1, j - 1]*(-1)^(n - j - i)*Binomial[n - i, j]/(n - i), {i, 0, n - k}, {j, k, n - i}]; r[n_, 0] = 1; Table[r[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 21 2013 *)
%Y A185812 Cf. A082395, apparently R(n,1), A097332 (row sums). - _R. J. Mathar_, Feb 10 2011
%K A185812 nonn,tabl
%O A185812 0,8
%A A185812 _Vladimir Kruchinin_, Feb 05 2011