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%I #35 Jul 29 2020 05:51:18
%S 1,2,1,5,4,1,13,14,6,1,34,46,27,8,1,89,145,107,44,10,1,233,444,393,
%T 204,65,12,1,610,1331,1371,854,345,90,14,1,1597,3926,4607,3336,1620,
%U 538,119,16,1,4181,11434,15045,12390,6997,2799,791,152,18,1
%N Riordan array (1, x*(1-x)/(1-3*x+x^2)).
%C The column of index 0 contains a 1 followed by zeros and is not reproduced in this triangle.
%C The second argument of the array definition is A(x) = A000045(x/(1-x)) = A001519(x)-1.
%C Triangle T(n,k), 1 <= k <= n, given by (0, 2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Jan 26 2012
%H Alois P. Heinz, <a href="/A188137/b188137.txt">Rows n = 1..141, flattened</a>
%H Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Janjic/janjic93.html">Words and Linear Recurrences</a>, J. Int. Seq. 21 (2018), #18.1.4.
%H Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F T(n,m) = Sum_{k=m..n} binomial(n-1,k-1) * Sum_{i=ceiling((k-m)/2)..k-m} binomial(i,k-m-i)*binomial(m+i-1,m-1), 0<m<=n.
%F T(n,m) = Sum_{i=1..n-m+1} A001519(i)*T(n-i,m-1).
%F T(n,1) = A001519(n).
%F Sum_{m=1..n} T(n,m) = A007052(n-1).
%F G.f.: (1-3x+x^2)/(1-(3+y)*x + (1+y)*x^2). - _Philippe Deléham_, Jan 26 2012
%e Triangle begins:
%e 1;
%e 2, 1;
%e 5, 4, 1;
%e 13, 14, 6, 1;
%e 34, 46, 27, 8, 1;
%e 89, 145, 107, 44, 10, 1;
%e From _Philippe Deléham_, Jan 26 2012: (Start)
%e Triangle (0,2,1/2,1/2,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,...) begins:
%e 1;
%e 0, 1;
%e 0, 2, 1;
%e 0, 5, 4, 1;
%e 0, 13, 14, 6, 1;
%e 0, 34, 46, 27, 8, 1;
%e 0, 89, 145, 107, 44, 10, 1; (End)
%p A188137 := proc(n,m) add( binomial(n-1,k-1) *add(binomial(i,k-m-i) *binomial(m+i-1,m-1),i=ceil((k-m)/2)..k-m),k=m..n) ; end proc:
%p seq(seq(A188137(n,k),k=1..n),n=1..10) ; # _R. J. Mathar_, Mar 30 2011
%t t[n_, m_] := Sum[ Binomial[n - 1, k - 1]*Sum[ Binomial[i, k - m - i]*Binomial[m + i - 1, m - 1], {i, Ceiling[(k - m)/2], k - m}], {k, m, n}]; Table[t[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Feb 21 2013, translated from Maxima *)
%o (Maxima)
%o T(n,m):=sum(binomial(n-1,k-1) *sum(binomial(i,k-m-i) *binomial(m+i-1,m-1), i,ceiling((k-m)/2),k-m), k,m,n);
%Y Cf. A001519 (column 1), A030267 (column 2).
%K nonn,tabl
%O 1,2
%A _Vladimir Kruchinin_, Mar 21 2011
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