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A132070 G.f. satisfies: A(x) = 1 + x*A(x) * A(x*A(x)) + x^2*A(x)^2 * A'(x*A(x)). 1
1, 1, 3, 16, 121, 1166, 13508, 181616, 2769085, 47109136, 883639920, 18107148225, 402374973506, 9637795235498, 247545497301618, 6787859682565805, 197933312666622029, 6116627434424525916, 199695744998600583128, 6868763166262716491823, 248281343442023732418598 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..20.

FORMULA

a(n) = A113882(n+1) / (n+1) for n>=0.

G.f. A(x) satisfies:

(1) A(x) = (1/x)*Series_Reversion(x/(1 + x*A(x) + x^2*A'(x))).

(2) A(x) = (1/x)*Series_Reversion(x/G(x)) = G(x*A(x)) where G(x) = 1 + x*A(x) + x^2*A'(x) = A(x/G(x)) is the g.f. of A113882 (number of well-nested drawings of a rooted tree).

EXAMPLE

G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 121*x^4 + 1166*x^5 + 13508*x^6 + ...

Related expansions:

A(x*A(x)) = 1 + x + 4*x^2 + 25*x^3 + 206*x^4 + 2077*x^5 + 24603*x^6 + ...

A'(x*A(x)) = 1 + 6*x + 54*x^2 + 598*x^3 + 7714*x^4 + 112726*x^5 + ...

A(x)*A(A(x)) = 1 + 2*x + 8*x^2 + 48*x^3 + 380*x^4 + 3709*x^5 + 42856*x^6 + ...

A(x)^2*A'(A(x)) = 1 + 8*x + 73*x^2 + 786*x^3 + 9799*x^4 + 138760*x^5 + ...

where A(x) = 1 + x*A(x)*A(x*A(x)) + x^2*A(x)^2*A'(x*A(x)).

Let G(x) = 1 + x*A(x) + x^2*A'(x) be the g.f. of A113882;

the table of coefficients in G(x)^n begins:

G(x)^1: [(1), 1, 2, 9, 64, 605, 6996, 94556, ...];

G(x)^2: [1, (2), 5, 22, 150, 1374, 15539, 206676, ...];

G(x)^3: [1, 3, (9), 40, 264, 2346, 25937, 339294, ...];

G(x)^4: [1, 4, 14, (64), 413, 3568, 38558, 495848, ...];

G(x)^5: [1, 5, 20, 95, (605), 5096, 53840, 680365, ...];

G(x)^6: [1, 6, 27, 134, 849, (6996), 72302, 897558, ...];

G(x)^7: [1, 7, 35, 182, 1155, 9345, (94556), 1152936, ...]; ...

where the terms in parentheses form the initial terms of this sequence:

[(1)/1, (2)/2, (9)/3, (64)/4, (605)/5, (6996)/6, (94556)/7, ...].

Further, the logarithm of the g.f. A(x) may be formed from a diagonal in the above table:

log(A(x)) = x + 5*x^2/2 + 40*x^3/3 + 413*x^4/4 + 5096*x^5/5 + 72302*x^6/6 + 1152936*x^7/7 + ...

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A*subst(A, x, x*A+x*O(x^n))+x^2*A^2*subst(A', x, x*A+x*O(x^n))); polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

(PARI) {a(n)=local(G=1+x); for(k=0, n, G=1+x*deriv(serreverse(x/(G+x^2*O(x^n))))); polcoeff(serreverse(x/G)/x, n)}

CROSSREFS

Cf. A113882.

Sequence in context: A003692 A166883 A145158 * A121629 A200793 A141625

Adjacent sequences:  A132067 A132068 A132069 * A132071 A132072 A132073

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 08 2007

EXTENSIONS

More terms and name changed by Paul D. Hanna, Apr 28 2012

STATUS

approved

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Last modified May 18 15:25 EDT 2021. Contains 343995 sequences. (Running on oeis4.)