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