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%I #2 Mar 30 2012 18:37:20
%S 1,1,2,7,33,187,1208,8626,66782,553355,4862938,45025668,437029462,
%T 4429568600,46738108168,512097036882,5814415004953,68293044651990,
%U 828547526906649,10369747261970151,133732024450930096
%N G.f. satisfies: A(x) = x + x*Sum_{n>=1} G_{n}(x)^n, where G_{n}(x) is the n-th iteration of A(x) defined by G{n}(x) = G_{n-1}(A(x)) with G_0(x)=x.
%e G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 33*x^5 + 187*x^6 +...
%e Related expansions:
%e A(A(x)) = x + 2*x^2 + 6*x^3 + 25*x^4 + 130*x^5 + 789*x^6 +...
%e A(A(x))^2 = x^2 + 4*x^3 + 16*x^4 + 74*x^5 + 396*x^6 +...
%e A(A(A(x))) = x + 3*x^2 + 12*x^3 + 60*x^4 + 353*x^5 + 2348*x^6 +...
%e A(A(A(x)))^3 = x^3 + 9*x^4 + 63*x^5 + 423*x^6 + 2895*x^7 +...
%e Coefficients in the n-th iteration of the g.f. begin:
%e n=1: [1, 1, 2, 7, 33, 187, 1208, 8626, 66782, 553355, ...];
%e n=2: [1, 2, 6, 25, 130, 789, 5376, 40140, 323198, 2775204, ...];
%e n=3: [1, 3, 12, 60, 353, 2348, 17208, 136574, 1159754, ...];
%e n=4: [1, 4, 20, 118, 788, 5800, 46236, 394382, 3568108, ...];
%e n=5: [1, 5, 30, 205, 1545, 12595, 109664, 1010914, 9803334, ...];
%e n=6: [1, 6, 42, 327, 2758, 24817, 235932, 2354038, 24532158, ...];
%e n=7: [1, 7, 56, 490, 4585, 45304, 469000, 5059950, 56677550, ...];
%e n=8: [1, 8, 72, 700, 7208, 77768, 873352, 10164212, 122210376, ...];
%e n=9: [1, 9, 90, 963, 10833, 126915, 1539720, 19271058, 248179134, ...];
%e n=10:[1, 10, 110, 1285, 15690, 198565, 2591528, 34766008, 478309118, ...]; ...
%e Coefficients in the n-th power of the n-th iteration of the g.f. begin:
%e n=1: [1, 1, 2, 7, 33, 187, 1208, 8626, 66782, 553355, ...];
%e n=2: [0, 1, 4, 16, 74, 396, 2398, 16093, 117752, 927818, ...];
%e n=3: [0, 0, 1, 9, 63, 423, 2895, 20634, 154323, 1213566, ...];
%e n=4: [0, 0, 0, 1, 16, 176, 1688, 15312, 136320, 1214472, ...];
%e n=5: [0, 0, 0, 0, 1, 25, 400, 5275, 62850, 707350, 7710070, ...];
%e n=6: [0, 0, 0, 0, 0, 1, 36, 792, 13842, 212028, 2989698, ...];
%e n=7: [0, 0, 0, 0, 0, 0, 1, 49, 1421, 31899, 614166, 10685675, ...];
%e n=8: [0, 0, 0, 0, 0, 0, 0, 1, 64, 2368, 66528, 1577280, ...];
%e n=9: [0, 0, 0, 0, 0, 0, 0, 0, 1, 81, 3726, 128223, 3676887, ...];
%e n=10:[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 100, 5600, 231850, 7917900, ...]; ...
%e Column sums in the above table form this sequence shift left 1 place.
%o (PARI) {a(n)=local(a_n=0,G=x,F=x+sum(k=2,n-1,a(k)*x^k)); if(n<1,0,if(n==1,1, for(k=1,n-1,G=x; for(i=1,k,G=subst(F,x,G+x*O(x^n)));a_n=a_n+polcoeff(G^k,n-1));a_n))}
%Y Cf. A171780 (variant).
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 03 2010
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