A185937 Riordan array (A000045(x)^m, x*A000108(x)), m = 1.

0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 5, 3, 1, 0, 5, 12, 9, 4, 1, 0, 8, 31, 26, 14, 5, 1, 0, 13, 85, 77, 46, 20, 6, 1, 0, 21, 248, 235, 150, 73, 27, 7, 1, 0, 34, 762, 741, 493, 258, 108, 35, 8, 1, 0, 55, 2440, 2406, 1644, 903, 410, 152, 44, 9, 1, 0

Offset

0,7

Comments

Essentially A139375 with zero diagonal added. - Ralf Stephan, Jan 01 2014

Links

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

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties , arXiv:1103.2582 [math.CO], 2013.

Formula

For m=1: R(n,k,m) = k*Sum_{i=0..n-k} (Sum_{j=ceiling((i-m)/2)..i-m} binomial(j, i-m-j) * binomial(m+j-1, m-1)) * binomial(2*(n-i)-k-1, n-i-1)/(n-i) if k > 0; R(n,0,m) = Sum_{j=ceiling((n-m)/2)..n-m} binomial(j, n-m-j) * binomial(m+j-1, m-1).

Example

Array begins
   0;
   1,  0;
   1,  1,  0;
   2,  2,  1,  0;
   3,  5,  3,  1,  0;
   5, 12,  9,  4,  1,  0;
   8, 31, 26, 14,  5,  1,  0;
  13, 85, 77, 46, 20,  6,  1,  0;

Mathematica

r[n_, k_, m_] := k*Sum[ Sum[ Binomial[j, i-m-j]*Binomial[m+j-1, m-1], {j, Ceiling[(i-m)/2], i-m}] * Binomial[2*(n-i)-k-1, n-i-1]/(n-i), {i, 0, n-k}]; r[n_, 0, m_] := Sum[ Binomial[j, n-m-j]*Binomial[m+j-1, m-1], {j, Ceiling[(n-m)/2], n-m}]; Table[r[n, k, 1], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 14 2013, after Vladimir Kruchinin *)

Crossrefs

Cf. A000045, A000108, A139375.

Sequence in context: A276543 A107424 A155161 * A292086 A065177 A064044

Adjacent sequences: A185934 A185935 A185936 * A185938 A185939 A185940

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Feb 06 2011

Status

approved