A185812 Riordan array ( 1/(1-x), x*A005043(x) ).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 6, 5, 4, 1, 1, 1, 12, 12, 7, 5, 1, 1, 1, 27, 26, 19, 9, 6, 1, 1, 1, 63, 63, 43, 27, 11, 7, 1, 1, 1, 154, 153, 110, 63, 36, 13, 8, 1, 1, 1, 386, 386, 275, 169, 86, 46, 15, 9, 1, 1

Offset: 0

Author

Vladimir Kruchinin, Feb 05 2011

Keywords

• nonn
• tabl

Examples

			Array begins:
  1;
  1,  1;
  1,  1,  1;
  1,  2,  1,  1;
  1,  3,  3,  1,  1;
  1,  6,  5,  4,  1,  1;
  1, 12, 12,  7,  5,  1,  1;
  1, 27, 26, 19,  9,  6,  1,  1;
		

Links

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

Crossrefs

Cf. A082395, apparently R(n,1), A097332 (row sums). - R. J. Mathar, Feb 10 2011

Programs

Maple

A185812 := proc(n,k) if n = k  or k =0 then 1; else k*add(1/(n-i)*add(binomial(2*j-k-1,j-1) *(-1)^(n-j-i) *binomial(n-i,j),j=k..n-i),i=0..n-k) ; end if; end proc:
seq(seq(A185812(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Feb 10 2011

Mathematica

r[n_, k_] := k*Sum[Binomial[2*j - k - 1, j - 1]*(-1)^(n - j - i)*Binomial[n - i, j]/(n - i), {i, 0, n - k}, {j, k, n - i}]; r[n_, 0] = 1; Table[r[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 21 2013 *)

Formula

R(n,k) = k*Sum_{i=0..(n-k)} (Sum_{j=k..(n-i)} binomial(2*j-k-1,j-1) *(-1)^(n-j-i) *binomial(n-i,j))/(n-i), k>0.

R(n,0)=1.