A185937 - OEIS

%I #37 Oct 13 2017 03:32:29

%S 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,

%T 46,20,6,1,0,21,248,235,150,73,27,7,1,0,34,762,741,493,258,108,35,8,1,

%U 0,55,2440,2406,1644,903,410,152,44,9,1,0

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

%C Essentially A139375 with zero diagonal added. - _Ralf Stephan_, Jan 01 2014

%H G. C. Greubel, <a href="/A185937/b185937.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H Vladimir Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties </a>, arXiv:1103.2582 [math.CO], 2013.

%F 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).

%e Array begins

%e    0;

%e    1,  0;

%e    1,  1,  0;

%e    2,  2,  1,  0;

%e    3,  5,  3,  1,  0;

%e    5, 12,  9,  4,  1,  0;

%e    8, 31, 26, 14,  5,  1,  0;

%e   13, 85, 77, 46, 20,  6,  1,  0;

%t 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_ *)

%Y Cf. A000045, A000108, A139375.

%K nonn,tabl

%O 0,7

%A _Vladimir Kruchinin_, Feb 06 2011