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

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, 8, 1, 36, 471, 443, 273, 124, 41, 9, 1, 45, 1398, 1362, 891, 448, 175, 51, 10, 1, 55, 4381, 4336, 2938, 1576, 685, 237, 62, 11, 1, 66, 14282, 14227, 9846, 5510, 2572, 996, 311, 74, 12, 1

Offset: 0

Author

Vladimir Kruchinin, Feb 07 2011

Keywords

• nonn
• tabl

Examples

			Array begins
   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,   8,  1;
  36, 471, 443, 273, 124,  41,  9,   1;
Production matrix begins:
   3, 1;
  -3, 1, 1;
   4, 1, 1, 1;
  -2, 1, 1, 1, 1;
   1, 1, 1, 1, 1, 1;
   0, 1, 1, 1, 1, 1, 1;
   0, 1, 1, 1, 1, 1, 1, 1;
   0, 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), A185945 (m=4), A014151 (column k=1).

Cf. A000108, A000217.

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, 3] = (n + 1)*(n + 2)/2; Table[ r[n, k, 3], {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=3, k>0.

R(n,0,3) = (n+1)*(n+2)/2 = A000217(n+1).