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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: A075248 A359140 A336707 * A307883 A111528 A144303

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 February 7 19:43 EST 2023. Contains 360128 sequences. (Running on oeis4.)