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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. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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: A145236 A162206 A075248 * A111528 A144303 A287024

Adjacent sequences:  A128322 A128323 A128324 * A128326 A128327 A128328

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Mar 11 2007

STATUS

approved

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Last modified August 15 09:12 EDT 2018. Contains 313756 sequences. (Running on oeis4.)