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%I #29 Oct 15 2018 20:17:47
%S 1,1,2,9,60,504,4946,54430,655362,8496454,117311198,1711459903,
%T 26228829200,420370445830,7021029571856,121859518887327,
%U 2192820745899978,40831103986939664,785429260324068156,15585831041632684997,318649154587152781210
%N G.f. satisfies: A(x/A(x)^2) = 1 + x ; thus A(x) = 1 + Series_Reversion(x/A(x)^2).
%C From _Paul D. Hanna_, Nov 16 2008: (Start)
%C More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then
%C A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);
%C thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1. (End)
%H Vaclav Kotesovec, <a href="/A120970/b120970.txt">Table of n, a(n) for n = 0..350</a>
%F G.f. satisfies: A(x) = 1 + x*A(A(x) - 1)^2.
%F Let B(x) be the g.f. of A120971, then B(x) and g.f. A(x) are related by:
%F (a) B(x) = A(A(x)-1),
%F (b) B(x) = A(x*B(x)^2),
%F (c) A(x) = B(x/A(x)^2),
%F (d) A(x) = 1 + x*B(x)^2,
%F (e) B(x) = 1 + x*B(x)^2*B(A(x)-1)^2,
%F (f) A(B(x)-1) = B(A(x)-1) = B(x*B(x)^2).
%F a(n) = [x^(n-1)] (1/n)*A(x)^(2n) for n>=1 with a(0)=1; i.e., a(n) equals 1/n times the coefficient of x^(n-1) in A(x)^(2n) for n>=1. [_Paul D. Hanna_, Nov 16 2008]
%e G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 60*x^4 + 504*x^5 + 4946*x^6 + ...
%e Related expansions.
%e A(x)^2 = 1 + 2*x + 5*x^2 + 22*x^3 + 142*x^4 + 1164*x^5 + 11221*x^6 + ...
%e A(A(x)-1) = 1 + x + 4*x^2 + 26*x^3 + 218*x^4 + 2151*x^5 + 23854*x^6 + ...
%e A(A(x)-1)^2 = 1 + 2*x + 9*x^2 + 60*x^3 + 504*x^4 + 4946*x^5 + ...
%e x/A(x)^2 = x - 2*x^2 - x^3 - 10*x^4 - 73*x^5 - 662*x^6 - 6842*x^7 - ...
%e Series_Reversion(x/A(x)^2) = x + 2*x^2 + 9*x^3 + 60*x^4 + 504*x^5 + 4946*x^6 + ...
%e To illustrate the formula a(n) = [x^(n-1)] 2*A(x)^(2*n)/(2*n),
%e form a table of coefficients in A(x)^(2*n) as follows:
%e A^2: [(1), 2, 5, 22, 142, 1164, 11221, 121848, ...];
%e A^4: [1, (4), 14, 64, 397, 3116, 29002, 306468, ...];
%e A^6: [1, 6, (27), 134, 825, 6270, 56492, 580902, ...];
%e A^8: [1, 8, 44, (240), 1502, 11200, 98144, 983016, ...];
%e A^10: [1, 10, 65, 390, (2520), 18672, 160115, 1565260, ...];
%e A^12: [1, 12, 90, 592, 3987, (29676), 250730, 2399388, ...];
%e A^14: [1, 14, 119, 854, 6027, 45458, (381010), 3582266, ...]; ...
%e in which the main diagonal forms the initial terms of this sequence:
%e [2/2*(1), 2/4*(4), 2/6*(27), 2/8*(240), 2/10*(2520), 2/12*(29676), ...].
%t terms = 21; A[_] = 1; Do[A[x_] = 1 + x*A[A[x] - 1]^2 + O[x]^j // Normal, {j, terms}]; CoefficientList[A[x], x] (* _Jean-François Alcover_, Jan 15 2018 *)
%o (PARI) {a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0); A[ #A]=-Vec(subst(Ser(A),x,x/Ser(A)^2))[ #A]);A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) /* This sequence is generated when k=2, m=0: A(x/A(x)^k) = 1 + x*A(x)^m */ {a(n,k=2,m=0)=local(A=sum(i=0,n-1,a(i,k,m)*x^i));if(n==0,1,polcoeff((m+k)/(m+k*n)*A^(m+k*n),n-1))} \\ _Paul D. Hanna_, Nov 16 2008
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A120971; variants: A120972, A120974, A120976, A030266, A067145, A107096.
%Y Cf. related variants: A145347, A145348, A147664, A145349, A145350. - _Paul D. Hanna_, Nov 16 2008
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 20 2006
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