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

Author

Paul D. Hanna, Mar 11 2007

Keywords

• nonn
• tabl

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.

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

Cf. A030266 (row 0), A128326 (row 1), A128327 (row 2), A128328 (row 3), A128329 (main diagonal); A128330 (variant).

Programs

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) ].