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

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, 301, 294, 190, 92, 33, 8, 1, 9, 927, 919, 618, 324, 134, 42, 9, 1, 10, 2983, 2974, 2047, 1128, 510, 186, 52, 10, 1, 11, 9901, 9891, 6908, 3934, 1887, 759, 249, 63, 11, 1

Offset: 0

Author

Vladimir Kruchinin, Feb 07 2011

Keywords

• nonn
• tabl

Examples

			Array begins
  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, 301, 294, 190,  92,  33,   8,   1;
Production matrix begins:
   2, 1;
  -1, 1, 1;
   1, 1, 1, 1;
   0, 1, 1, 1, 1;
   0, 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), A185944 (m=3), A185945 (m=4).

Row sums A014140. Cf. A000108, A014143.

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, 2] := n + 1; Table[r[n, k, 2], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 13 2012, from formula *)

Sage

@CachedFunction
def A(n, k):
    if n==k: return n+1
    return add(A(n-1, j) for j in (0..k))
A185943 = lambda n,k: A(n, n-k)
for n in (0..7) :
     print([A185943(n, k) for k in (0..n)])  # Peter Luschny, Nov 14 2012

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 = 2, k > 0.

R(n,0,2) = n + 1.

Conjecture: R(n,1,2) = A014140(n-1). R(n,2,2) = A014143(n-2). - R. J. Mathar, Feb 11 2011