OFFSET
1,2
COMMENTS
T(n,1)/n is A000260(n). - Vladimir Kruchinin, Apr 02 2015
LINKS
Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
FORMULA
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
EXAMPLE
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+
...
PROG
(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 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 */
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, Mar 20 2011
STATUS
approved