A128325 Rectangular table, read by antidiagonals, where the g.f.s of row n, R(x,n), satisfy: R(x,n+1) = R(G(x),n) for n>=0 and x*R(x,0) = G(x) = x + x*G(G(x)) is the g.f. of A030266.
1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 12, 23, 1, 1, 5, 20, 57, 104, 1, 1, 6, 30, 114, 305, 531, 1, 1, 7, 42, 200, 712, 1787, 2982, 1, 1, 8, 56, 321, 1435, 4772, 11269, 18109, 1, 1, 9, 72, 483, 2608, 10900, 33896, 75629, 117545, 1, 1, 10, 90, 692, 4389, 22219, 86799
Offset: 0
Examples
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, ...]; ...
Crossrefs
Programs
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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]}
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) ].
Comments