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A188111 Triangle T(n,m) read by rows, [A(x)]^m = Sum_{n>=m} T(n,m)*x^n, where A(x) satisfies A(x) = x/(1-A(x)-A(x)^2).

%I A188111 #35 Oct 19 2024 21:27:17
%S A188111 1,1,1,3,2,1,10,7,3,1,38,26,12,4,1,154,105,49,18,5,1,654,444,210,80,
%T A188111 25,6,1,2871,1944,927,363,120,33,7,1,12925,8734,4191,1672,575,170,42,
%U A188111 8,1,59345,40040,19305,7810,2761,858,231,52,9,1
%N A188111 Triangle T(n,m) read by rows, [A(x)]^m = Sum_{n>=m} T(n,m)*x^n, where A(x) satisfies A(x) = x/(1-A(x)-A(x)^2).
%H A188111 Vladimir Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F A188111 T(n,m) = A037027(2*n-m-1,n-1)*m/n.
%F A188111 T(n,m) = Sum_{i=1..n-m+1} A001002(i-1)*T(n-i,m-1), m>0. T(n,1) = A001002(n-1).
%F A188111 T(n,m) = if n<0 or m<0 or n<m then 0 else if n=m then 1 else if m=0 then 0 else T(n-1,m-1)+T(n,m+1)+T(n,m+2). - _Vladimir Kruchinin_, Apr 21 2016
%F A188111 Assuming the range n>=0 and 0<=k<=n we have T(n,k) = C(2*n-k,n)*hypergeom([(k-n)/2, (k-n+1)/2], [k-2*n],-4)*(k+1)/(n+1) for n>=1 and T(0,0) = 1. - _Peter Luschny_, Apr 25 2016
%e A188111 Triangle starts:
%e A188111     1;
%e A188111     1,   1;
%e A188111     3,   2,  1;
%e A188111    10,   7,  3,  1;
%e A188111    38,  26, 12,  4, 1;
%e A188111   154, 105, 49, 18, 5, 1;
%e A188111   ...
%p A188111 T := (n,k) -> `if`(n=0,1,binomial(2*n-k,n)*hypergeom([(k-n)/2, (k-n+1)/2], [k-2*n],-4)*(k+1)/(n+1)): seq(seq(simplify(T(n,k), k=0..n)), n=0..10); # _Peter Luschny_, Apr 25 2016
%t A188111 T[n_, m_] := T[n, m] = Which[n <= 0 || m <= 0, 0, n < m, 0, n == m, 1, True, T[n-1, m-1] + T[n, m+1] + T[n, m+2]];
%t A188111 Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 03 2019, after _Vladimir Kruchinin_ *)
%o A188111 (Maxima)
%o A188111 T(n, m):=if n<=0 or m<=0 then 0 else if n<m then 0 else if n=m then 1 else T(n-1, m-1)+T(n, m+1)+T(n, m+2); /* _Vladimir Kruchinin_, Apr 21 2016 */
%Y A188111 Cf. A001002, A037027.
%K A188111 nonn,tabl
%O A188111 1,4
%A A188111 _Vladimir Kruchinin_, Mar 21 2011