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A185943 Riordan array ((1/(1-x))^m, x*A000108(x)), m = 2.

%I A185943 #30 Mar 07 2020 14:54:45
%S A185943 1,2,1,3,3,1,4,7,4,1,5,16,12,5,1,6,39,34,18,6,1,7,104,98,59,25,7,1,8,
%T A185943 301,294,190,92,33,8,1,9,927,919,618,324,134,42,9,1,10,2983,2974,2047,
%U A185943 1128,510,186,52,10,1,11,9901,9891,6908,3934,1887,759,249,63,11,1
%N A185943 Riordan array ((1/(1-x))^m, x*A000108(x)), m = 2.
%H A185943 G. C. Greubel, <a href="/A185943/b185943.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185943 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 = 2, k > 0.
%F A185943 R(n,0,2) = n + 1.
%F A185943 Conjecture: R(n,1,2) = A014140(n-1). R(n,2,2) = A014143(n-2). - _R. J. Mathar_, Feb 11 2011
%e A185943 Array begins
%e A185943   1;
%e A185943   2,   1;
%e A185943   3,   3,   1;
%e A185943   4,   7,   4,   1;
%e A185943   5,  16,  12,   5,   1;
%e A185943   6,  39,  34,  18,   6,   1;
%e A185943   7, 104,  98,  59,  25,   7,   1;
%e A185943   8, 301, 294, 190,  92,  33,   8,   1;
%e A185943 Production matrix begins:
%e A185943    2, 1;
%e A185943   -1, 1, 1;
%e A185943    1, 1, 1, 1;
%e A185943    0, 1, 1, 1, 1;
%e A185943    0, 1, 1, 1, 1, 1;
%e A185943    0, 1, 1, 1, 1, 1, 1;
%e A185943    0, 1, 1, 1, 1, 1, 1, 1;
%e A185943    0, 1, 1, 1, 1, 1, 1, 1, 1;
%e A185943    ... _Philippe Deléham_, Sep 20 2014
%t A185943 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, 2] := n + 1; Table[r[n, k, 2], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 13 2012, from formula *)
%o A185943 (Sage)
%o A185943 @CachedFunction
%o A185943 def A(n, k):
%o A185943     if n==k: return n+1
%o A185943     return add(A(n-1, j) for j in (0..k))
%o A185943 A185943 = lambda n,k: A(n, n-k)
%o A185943 for n in (0..7) :
%o A185943      print([A185943(n, k) for k in (0..n)])  # _Peter Luschny_, Nov 14 2012
%Y A185943 Cf. A091491 (m=1), A185944 (m=3), A185945 (m=4).
%Y A185943 Row sums A014140. Cf. A000108, A014143.
%K A185943 nonn,tabl
%O A185943 0,2
%A A185943 _Vladimir Kruchinin_, Feb 07 2011