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

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, 40, 9, 1, 120, 750, 666, 387, 164, 50, 10, 1, 165, 2148, 2028, 1278, 612, 225, 61, 11, 1, 220, 6529, 6364, 4216, 2188, 910, 298, 73, 12, 1, 286, 20811, 20591, 14062, 7698, 3482, 1294, 384, 86, 13, 1

Offset: 0

Author

Vladimir Kruchinin, Feb 07 2011

Keywords

• nonn
• tabl

Examples

			Array begins
    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,  40,   9,   1;
  120, 750, 666, 387, 164,  50,  10,   1;
Production matrix begins:
   4, 1;
  -6, 1, 1;
  10, 1, 1, 1;
  -9, 1, 1, 1, 1;
   7, 1, 1, 1, 1, 1;
  -3, 1, 1, 1, 1, 1, 1;
   1, 1, 1, 1, 1, 1, 1, 1;
   0, 1, 1, 1, 1, 1, 1, 1, 1;
   0, 1, 1, 1, 1, 1, 1, 1, 1, 1;
   0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  ... _Philippe Deléham_, Sep 20 2014
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A091491 (m=1), A185943 (m=2), A185944 (m=3).

Cf. A000108, A000292.

Programs

Mathematica

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 *)

Formula

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.

R(n,0,4) = binomial(n+3,3) = A000292(n+1).