Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #9 Nov 12 2014 02:08:05
%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