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%I #6 Dec 20 2014 22:33:37
%S 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,
%T 1,1,7,42,200,712,1787,2982,1,1,8,56,321,1435,4772,11269,18109,1,1,9,
%U 72,483,2608,10900,33896,75629,117545,1,1,10,90,692,4389,22219,86799
%N 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.
%C 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.
%F Let R(x,n) denote the g.f. of row n of this table, then
%F R(x,n) = 1 + x*Product_{k=0..n+1} R(x,k),
%F R(x,n) = 1 + x/[1 - x*Sum_{k=1..n+2} R(x,k) ].
%e Consider the infinite system of simultaneous equations:
%e A = 1 + x*A*B;
%e B = 1 + x*A*B*C;
%e C = 1 + x*A*B*C*D;
%e D = 1 + x*A*B*C*D*E;
%e E = 1 + x*A*B*C*D*E*F; ...
%e The unique solution to the variables are:
%e A = R(x,0), B = R(x,1), C = R(x,2), D = R(x,3), E = R(x,4), etc.,
%e where R(x,n) denotes the g.f. of row n of this table and satisfies:
%e R(x,1) = R(x*A,0); R(x,2) = R(x*A,1); R(x,3) = R(x*A,2); etc.
%e The row g.f.s are also related by:
%e R(x,0) = 1 + x/(1 - x*R(x,1) - x*R(x,2));
%e R(x,1) = 1 + x/(1 - x*R(x,1) - x*R(x,2) - x*R(x,3));
%e R(x,2) = 1 + x/(1 - x*R(x,1) - x*R(x,2) - x*R(x,3) - x*R(x,4)); etc.
%e The initial rows of this table begin:
%e R(x,0): [1, 1, 2, 6, 23, 104, 531, 2982, 18109, ...];
%e R(x,1): [1, 1, 3, 12, 57, 305, 1787, 11269, 75629, ...];
%e R(x,2): [1, 1, 4, 20, 114, 712, 4772, 33896, 253102, ...];
%e R(x,3): [1, 1, 5, 30, 200, 1435, 10900, 86799, 720074, ...];
%e R(x,4): [1, 1, 6, 42, 321, 2608, 22219, 196910, 1805899, ...];
%e R(x,5): [1, 1, 7, 56, 483, 4389, 41531, 406441, 4095749, ...];
%e R(x,6): [1, 1, 8, 72, 692, 6960, 72512, 777888, 8559852, ...];
%e R(x,7): [1, 1, 9, 90, 954, 10527, 119832, 1399755, 16720998, ...];
%e R(x,8): [1, 1, 10, 110, 1275, 15320, 189275, 2392998, 30865353, ...];
%e R(x,9): [1, 1, 11, 132, 1661, 21593, 287859, 3918189, 54301621, ...];
%e R(x,10):[1, 1, 12, 156, 2118, 29624, 423956, 6183400, 91673594, ...]; ...
%o (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]}
%Y Cf. A030266 (row 0), A128326 (row 1), A128327 (row 2), A128328 (row 3), A128329 (main diagonal); A128330 (variant).
%K nonn,tabl
%O 0,6
%A _Paul D. Hanna_, Mar 11 2007
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