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)).

1, 2, 1, 9, 4, 1, 52, 22, 6, 1, 340, 140, 39, 8, 1, 2394, 969, 272, 60, 10, 1, 17710, 7084, 1995, 456, 85, 12, 1, 135720, 53820, 15180, 3542, 700, 114, 14, 1, 1068012, 420732, 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

Offset: 1

Author

Vladimir Kruchinin, Mar 20 2011

Keywords

• nonn
• tabl

Comments

T(n,1)/n is A000260(n). - Vladimir Kruchinin, Apr 02 2015

Examples

			1;
2, 1;
9, 4, 1;
52, 22, 6, 1;
340, 140, 39, 8, 1;
2394, 969, 272, 60, 10, 1;
17710, 7084, 1995, 456, 85, 12,...
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
taylor(1/(1-A(x)*y)-1,x,0,7,y,0,7);
(y)*x+
(2*y+y^2)*x^2+
(9*y+4*y^2+y^3)*x^3+
(52*y+22*y^2+6*y^3+y^4)*x^4+
(340*y+140*y^2+39*y^3+8*y^4+y^5)*x^5
(2394*y+969*y^2+272*y^3+60*y^4+10*y^5+y^6)*x^6+
(17710*y+7084*y^2+1995*y^3+456*y^4+85*y^5+12*y^6+y^7)*x^7+
...
		

Links

• Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Crossrefs

Cf. A000260, A039598, A069271.

Programs

Maxima

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 */
T(n,k):=if n<0 or k<0 or nVladimir Kruchinin, May 02 2015 */

Formula

T(n,m) = Sum_{i=1..n-m+1} A069271(i)*T(n-i,m-1), m>0, T(n,1) = A069271(n-1).

T(n,m) = (m/n)*A039598(2*n-m-1,n-1).

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

G.f.: 1/(1-x*y*A(x))-1, where A(x) is g.f. for A069271. - Vladimir Kruchinin, May 03 2015