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A185944 Riordan array ( (1/(1-x))^m , x*A000108(x) ), m = 3.

%I A185944 #23 Jul 23 2017 21:43:39
%S A185944 1,3,1,6,4,1,10,11,5,1,15,27,17,6,1,21,66,51,24,7,1,28,170,149,83,32,
%T A185944 8,1,36,471,443,273,124,41,9,1,45,1398,1362,891,448,175,51,10,1,55,
%U A185944 4381,4336,2938,1576,685,237,62,11,1,66,14282,14227,9846,5510,2572,996,311,74,12,1
%N A185944 Riordan array ( (1/(1-x))^m , x*A000108(x) ), m = 3.
%H A185944 G. C. Greubel, <a href="/A185944/b185944.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185944 R(n,k,m) = k*Sum_(i=0..n-k,binomial(i+m-1,m-1)*binomial(2*(n-i)-k-1,n-i-1)/(n-i)), m=3, k>0.
%F A185944 R(n,0,3) = (n+1)*(n+2)/2 = A000217(n+1).
%e A185944 Array begins
%e A185944    1;
%e A185944    3,   1;
%e A185944    6,   4,   1;
%e A185944   10,  11,   5,   1;
%e A185944   15,  27,  17,   6,   1;
%e A185944   21,  66,  51,  24,   7,   1;
%e A185944   28, 170, 149,  83,  32,   8,  1;
%e A185944   36, 471, 443, 273, 124,  41,  9,   1;
%e A185944 Production matrix begins:
%e A185944    3, 1;
%e A185944   -3, 1, 1;
%e A185944    4, 1, 1, 1;
%e A185944   -2, 1, 1, 1, 1;
%e A185944    1, 1, 1, 1, 1, 1;
%e A185944    0, 1, 1, 1, 1, 1, 1;
%e A185944    0, 1, 1, 1, 1, 1, 1, 1;
%e A185944    0, 1, 1, 1, 1, 1, 1, 1, 1;
%e A185944    ... _Philippe Deléham_, Sep 20 2014
%t A185944 r[n_, k_, m_] := k*Sum[ Binomial[i + m - 1, m - 1]*Binomial[2*(n - i) - k - 1, n - i - 1]/(n - i), {i, 0, n - k}]; r[n_, 0, 3] = (n + 1)*(n + 2)/2; Table[ r[n, k, 3], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 21 2013 *)
%Y A185944 Cf. A091491 (m=1), A185943 (m=2), A185945 (m=4), A014151 (column k=1).
%Y A185944 Cf. A000108, A000217.
%K A185944 nonn,tabl
%O A185944 0,2
%A A185944 _Vladimir Kruchinin_, Feb 07 2011