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

0,6

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.

Links

Table of n, a(n) for n=0..62.

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

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

Sequence in context: A365623 A336707 A383341 * A307883 A111528 A363007

Adjacent sequences: A128322 A128323 A128324 * A128326 A128327 A128328

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 11 2007

Status

approved