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A185945 Riordan array ( (1/(1-x))^m , x*A000108(x) ), m =4.

%I A185945 #23 Jul 23 2017 21:43:50
%S A185945 1,4,1,10,5,1,20,16,6,1,35,43,23,7,1,56,109,74,31,8,1,84,279,223,114,
%T A185945 40,9,1,120,750,666,387,164,50,10,1,165,2148,2028,1278,612,225,61,11,
%U A185945 1,220,6529,6364,4216,2188,910,298,73,12,1,286,20811,20591,14062,7698,3482,1294,384,86,13,1
%N A185945 Riordan array ( (1/(1-x))^m , x*A000108(x) ), m =4.
%H A185945 G. C. Greubel, <a href="/A185945/b185945.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185945 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=4, k > 0.
%F A185945 R(n,0,4) = binomial(n+3,3) = A000292(n+1).
%e A185945 Array begins
%e A185945     1;
%e A185945     4,   1;
%e A185945    10,   5,   1;
%e A185945    20,  16,   6,   1;
%e A185945    35,  43,  23,   7,   1;
%e A185945    56, 109,  74,  31,   8,   1;
%e A185945    84, 279, 223, 114,  40,   9,   1;
%e A185945   120, 750, 666, 387, 164,  50,  10,   1;
%e A185945 Production matrix begins:
%e A185945    4, 1;
%e A185945   -6, 1, 1;
%e A185945   10, 1, 1, 1;
%e A185945   -9, 1, 1, 1, 1;
%e A185945    7, 1, 1, 1, 1, 1;
%e A185945   -3, 1, 1, 1, 1, 1, 1;
%e A185945    1, 1, 1, 1, 1, 1, 1, 1;
%e A185945    0, 1, 1, 1, 1, 1, 1, 1, 1;
%e A185945    0, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A185945    0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A185945   ... _Philippe Deléham_, Sep 20 2014
%t A185945 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, 4] = Binomial[n + 3, 3]; Table[ r[n, k, 4], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 21 2013 *)
%Y A185945 Cf. A091491 (m=1), A185943 (m=2), A185944 (m=3).
%Y A185945 Cf. A000108, A000292.
%K A185945 nonn,tabl
%O A185945 0,2
%A A185945 _Vladimir Kruchinin_, Feb 07 2011