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 A249791 Given g.f. A(x), let B(x) = 1 + x*A(x)^2 and C(x) = 1 + x*A(x)^3, then B(x*C(x)) = C(x) and C(x/B(x)) = B(x). 1

%I

%S 1,1,5,44,530,7911,139129,2798844,63178500,1578855377,43245568061,

%T 1288116498182,41457303331745,1433966498431138,53058288363011906,

%U 2091593330699875406,87527301512425974261,3875685191976323542974,181061755084572933223563,8900849566241379829936126

%N Given g.f. A(x), let B(x) = 1 + x*A(x)^2 and C(x) = 1 + x*A(x)^3, then B(x*C(x)) = C(x) and C(x/B(x)) = B(x).

%H Paul D. Hanna, <a href="/A249791/b249791.txt">Table of n, a(n) for n = 0..200</a>

%F G.f. A(x) satisfies:

%F (1) A( x*(1 + x*A(x)^3) )^2 * (1 + x*A(x)^3) = A(x)^3.

%F (2) A( x/(1 + x*A(x)^2) )^3 / (1 + x*A(x)^2) = A(x)^2.

%F (3) x / Series_Reversion( x*(1 + x*A(x)^3) ) = 1 + x*A(x)^2.

%F (4) (1/x) * Series_Reversion( x/(1 + x*A(x)^2) ) = 1 + x*A(x)^3.

%e G.f.: A(x) = 1 + x + 5*x^2 + 44*x^3 + 530*x^4 + 7911*x^5 + 139129*x^6 +...

%e RELATED SERIES.

%e A(x)^2 = 1 + 2*x + 11*x^2 + 98*x^3 + 1173*x^4 + 17322*x^5 + 301316*x^6 +...

%e A(x)^3 = 1 + 3*x + 18*x^2 + 163*x^3 + 1944*x^4 + 28440*x^5 + 489596*x^6 +...

%e A(x)^4 = 1 + 4*x + 26*x^2 + 240*x^3 + 2859*x^4 + 41492*x^5 + 707330*x^6 +...

%e A(x)^2 + x*A(x)^4 = 1 + 3*x + 15*x^2 + 124*x^3 + 1413*x^4 + 20181*x^5 +...

%e A(x)^3/(1 + x*A(x)^3) = 1 + 2*x + 13*x^2 + 126*x^3 + 1580*x^4 + 23978*x^5 +...

%e A( x*(1 + x*A(x)^3) )^2 = 1 + 2*x + 13*x^2 + 126*x^3 + 1580*x^4 + 23978*x^5 +...

%e A( x/(1 + x*A(x)^2) )^3 = 1 + 3*x + 15*x^2 + 124*x^3 + 1413*x^4 + 20181*x^5 +...

%e A( x*(1 + x*A(x)^3) ) = 1 + x + 6*x^2 + 57*x^3 + 715*x^4 + 10932*x^5 +...

%e A( x/(1 + x*A(x)^2) ) = 1 + x + 4*x^2 + 33*x^3 + 385*x^4 + 5644*x^5 +...

%e The table of coefficients in (1 + x*A(x)^2)^n begins:

%e n=1: [1, 1, 2, 11, 98, 1173, 17322, 301316, 6001696, ...];

%e n=2: [1, 2, 5, 26, 222, 2586, 37503, 644124, 12710722, ...];

%e n=3: [1, 3, 9, 46, 378, 4284, 60977, 1033614, 20201490, ...];

%e n=4: [1, 4, 14, 72, 573, 6320, 88246, 1475664, 28556726, ...];

%e n=5: [1, 5, 20, 105, 815, 8756, 119890, 1976935, 37868480, ...];

%e n=6: [1, 6, 27, 146, 1113, 11664, 156578, 2544978, 48239262, ...];

%e n=7: [1, 7, 35, 196, 1477, 15127, 199080, 3188354, 59783318, ...];

%e n=8: [1, 8, 44, 256, 1918, 19240, 248280, 3916768, 72628061, ...];

%e n=9: [1, 9, 54, 327, 2448, 24111, 305190, 4741218, 86915673, ...]; ...

%e in which the main diagonal generates coefficients in (1 + x*A(x)^3):

%e [1, 2/2, 9/3, 72/4, 815/5, 11664/6, 199080/7, 3916768/8, 86915673/9, ...]

%e = [1, 1, 3, 18, 163, 1944, 28440, 489596, 9657297, ...].

%o (PARI) {a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,0);

%o A[#A]=Vec(serreverse(x/(1+x*Ser(A)^2))/x - x*Ser(A)^3)[#A+1]);A[n+1]}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,0);

%o A[#A]=-Vec(x/serreverse(x*(1+x*Ser(A)^3)) - x*Ser(A)^2)[#A+1]);A[n+1]}

%o for(n=0,30,print1(a(n),", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 11 2014

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Last modified December 4 04:22 EST 2020. Contains 338921 sequences. (Running on oeis4.)