OFFSET
0,6
COMMENTS
Row n equals 1 + (n+2)-th self-composition of the g.f. G(x) of A030266: R(x,0) = 1 + G(G(x); R(x,1) = 1 + G(G(G(x))); R(x,2) = 1 + G(G(G(G(x)))); etc.
FORMULA
Let R(x,n) denote the g.f. of row n of this table, then
R(x,n) = 1 + x*Product_{k=0..n+1} R(x,k),
R(x,n) = 1 + x/[1 - x*Sum_{k=1..n+2} R(x,k) ].
EXAMPLE
Consider the infinite system of simultaneous equations:
A = 1 + x*A*B;
B = 1 + x*A*B*C;
C = 1 + x*A*B*C*D;
D = 1 + x*A*B*C*D*E;
E = 1 + x*A*B*C*D*E*F; ...
The unique solution to the variables are:
A = R(x,0), B = R(x,1), C = R(x,2), D = R(x,3), E = R(x,4), etc.,
where R(x,n) denotes the g.f. of row n of this table and satisfies:
R(x,1) = R(x*A,0); R(x,2) = R(x*A,1); R(x,3) = R(x*A,2); etc.
The row g.f.s are also related by:
R(x,0) = 1 + x/(1 - x*R(x,1) - x*R(x,2));
R(x,1) = 1 + x/(1 - x*R(x,1) - x*R(x,2) - x*R(x,3));
R(x,2) = 1 + x/(1 - x*R(x,1) - x*R(x,2) - x*R(x,3) - x*R(x,4)); etc.
The initial rows of this table begin:
R(x,0): [1, 1, 2, 6, 23, 104, 531, 2982, 18109, ...];
R(x,1): [1, 1, 3, 12, 57, 305, 1787, 11269, 75629, ...];
R(x,2): [1, 1, 4, 20, 114, 712, 4772, 33896, 253102, ...];
R(x,3): [1, 1, 5, 30, 200, 1435, 10900, 86799, 720074, ...];
R(x,4): [1, 1, 6, 42, 321, 2608, 22219, 196910, 1805899, ...];
R(x,5): [1, 1, 7, 56, 483, 4389, 41531, 406441, 4095749, ...];
R(x,6): [1, 1, 8, 72, 692, 6960, 72512, 777888, 8559852, ...];
R(x,7): [1, 1, 9, 90, 954, 10527, 119832, 1399755, 16720998, ...];
R(x,8): [1, 1, 10, 110, 1275, 15320, 189275, 2392998, 30865353, ...];
R(x,9): [1, 1, 11, 132, 1661, 21593, 287859, 3918189, 54301621, ...];
R(x,10):[1, 1, 12, 156, 2118, 29624, 423956, 6183400, 91673594, ...]; ...
PROG
(PARI) {T(n, k)=local(A=vector(n+k+3, m, 1+x+x*O(x^(n+k)))); for(i=1, n+k+3, for(j=1, n+k+1, N=n+k+2-j; A[N]=1+x/(1-x*sum(m=2, N+2, A[m]+x*O(x^(n+k)))))); Vec(A[n+1])[k+1]}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Mar 11 2007
STATUS
approved