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A188108 Triangle T(n,m) read by rows, obtained from [A(x)]^m = Sum_{n>=m} T(n,m)*x^n, where A(x) (the g.f. for A069271) satisfies 2*x^2*A(x)^3 = 1 - 2*x*A(x) - sqrt(1-4*x*A(x)).

%I A188108 #34 May 03 2016 12:45:10
%S A188108 1,2,1,9,4,1,52,22,6,1,340,140,39,8,1,2394,969,272,60,10,1,17710,7084,
%T A188108 1995,456,85,12,1,135720,53820,15180,3542,700,114,14,1,1068012,420732,
%U A188108 118755,28080,5750,1012,147,16,1,8579560,3362260,949344,226548,47502,8775,1400,184,18,1,70068713,27343888,7721604,1855040,395560,75516,12789,1872,225,20,1,580034052,225568798,63698830,15380937,3321120,649264,114576,17980,2436,270,22,1
%N A188108 Triangle T(n,m) read by rows, obtained from [A(x)]^m = Sum_{n>=m} T(n,m)*x^n, where A(x) (the g.f. for A069271) satisfies 2*x^2*A(x)^3 = 1 - 2*x*A(x) - sqrt(1-4*x*A(x)).
%C A188108 T(n,1)/n is A000260(n). - _Vladimir Kruchinin_, Apr 02 2015
%H A188108 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 A188108 T(n,m) = Sum_{i=1..n-m+1} A069271(i)*T(n-i,m-1), m>0, T(n,1) = A069271(n-1).
%F A188108 T(n,m) = (m/n)*A039598(2*n-m-1,n-1).
%F A188108 T(n,m) = Sum_{k=1..n-m} ((2*m*binomial(2*m+2*k-1,k))/(2*m+k)*T(n-m,k)), T(n,n)=1. - _Vladimir Kruchinin_, Apr 02 2015
%F A188108 G.f.: 1/(1-x*y*A(x))-1, where A(x) is g.f. for A069271.   - _Vladimir Kruchinin_, May 03 2015
%e A188108 1;
%e A188108 2, 1;
%e A188108 9, 4, 1;
%e A188108 52, 22, 6, 1;
%e A188108 340, 140, 39, 8, 1;
%e A188108 2394, 969, 272, 60, 10, 1;
%e A188108 17710, 7084, 1995, 456, 85, 12,...
%e A188108 A(x)=x+2*x^2+9*x^3+52*x^4+340*x^5+2394*x^6+17710*x^7+135720*x^8+1068012*x^9+8579560*x^10
%e A188108 taylor(1/(1-A(x)*y)-1,x,0,7,y,0,7);
%e A188108 (y)*x+
%e A188108 (2*y+y^2)*x^2+
%e A188108 (9*y+4*y^2+y^3)*x^3+
%e A188108 (52*y+22*y^2+6*y^3+y^4)*x^4+
%e A188108 (340*y+140*y^2+39*y^3+8*y^4+y^5)*x^5
%e A188108 (2394*y+969*y^2+272*y^3+60*y^4+10*y^5+y^6)*x^6+
%e A188108 (17710*y+7084*y^2+1995*y^3+456*y^4+85*y^5+12*y^6+y^7)*x^7+
%e A188108 ...
%o A188108 (Maxima)
%o A188108 T(n,m):=if n=m then 1 else sum((2*m*binomial(2*m+2*k-1,k))/(2*m+k)*T(n-m,k),k,1,n-m); /* _Vladimir Kruchinin_, Apr 02 2015 */
%o A188108 T(n,k):=if n<0 or k<0 or n<k then 0 else if n=k then 1 else if k=0 or n=0 then 0  else T(n-1,k-1)+2*T(n,k+1)+T(n+1,k+3); /*  _Vladimir Kruchinin_, May 02 2015 */
%Y A188108 Cf. A000260, A039598, A069271.
%K A188108 nonn,tabl
%O A188108 1,2
%A A188108 _Vladimir Kruchinin_, Mar 20 2011