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A119819 a(n) equals the coefficient of x^(n-1) in the (n-1)-th iteration of g.f. A(x) for n>1, with a(1)=1.

%I A119819 #7 Aug 27 2013 01:59:55
%S A119819 1,1,2,12,138,2370,54190,1553258,53883088,2211883428,105760271082,
%T A119819 5819880201432,364979361177134,25865387272507770,2056021496464455000,
%U A119819 182094050389241652004,17861355920109599058260,1929874166854161381238676,228564755268775651632722308,29540844190975459101114949972
%N A119819 a(n) equals the coefficient of x^(n-1) in the (n-1)-th iteration of g.f. A(x) for n>1, with a(1)=1.
%C A119819 Here the zeroth iteration of A(x) equals x, the first iteration is itself, the 2nd iteration of A(x) = A(A(x)), etc.
%e A119819 The coefficients in the n-th iteration of g.f. A(x) begin:
%e A119819 n=1: [1, 1,  2,   12,   138,   2370,   54190,  1553258,   53883088, ...];
%e A119819 n=2: [1, 2,  6,   35,   370,   6000,  132344,  3704032,  126318024, ...];
%e A119819 n=3: [1, 3, 12,   75,   758,  11612,  245746,  6688885,  223699238, ...];
%e A119819 n=4: [1, 4, 20,  138,  1388,  20322,  411708, 10854152,  354952262, ...];
%e A119819 n=5: [1, 5, 30,  230,  2370,  33760,  656414, 16711414,  532707614, ...];
%e A119819 n=6: [1, 6, 42,  357,  3838,  54190, 1018484, 25016120,  775036254, ...];
%e A119819 n=7: [1, 7, 56,  525,  5950,  84630, 1553258, 36874397, 1107956996, ...];
%e A119819 n=8: [1, 8, 72,  740,  8888, 128972, 2337800, 53883088, 1568966580, ...];
%e A119819 n=9: [1, 9, 90, 1008, 12858, 192102, 3476622, 78308058, 2211883428, ...]; ...
%e A119819 where the diagonal of coefficients equals this sequence shift left 1 place.
%e A119819 ...
%e A119819 More explicitly, the successive iterations of g.f. A(x) begin:
%e A119819 A(x) = (1)x + x^2 + 2x^3 + 12x^4 + 138x^5 + 2370x^6 + 54190x^7 +...
%e A119819 A(A(x)) = x + (2)x^2 + 6x^3 + 35x^4 + 370x^5 + 6000x^6 + 132344x^7 +...
%e A119819 A(A(A(x))) = x + 3x^2 + (12)x^3 + 75x^4 + 758x^5 + 11612x^6 +...
%e A119819 A(A(A(A(x)))) = x + 4x^2 + 20x^3 + (138)x^4 + 1388x^5 + 20322x^6 +...
%e A119819 A(A(A(A(A(x))))) = x + 5x^2 + 30x^3 + 230x^4 + (2370)x^5 + 33760x^6+...
%e A119819 A(A(A(A(A(A(x)))))) = x + 6x^2 +42x^3 +357x^4 +3838x^5 + (54190)x^6+...
%e A119819 ...
%o A119819 (PARI) {a(n)=local(F=x+x^2+sum(m=3,n-1,a(m)*x^m), G=x+x*O(x^n)); if(n<1, 0,if(n<=2,1,for(i=1, n-1, G=subst(F, x, G)); return(polcoeff(G, n-1, x))))}
%o A119819 for(n=1,30,print1(a(n),", "))
%Y A119819 Cf. A112317, A119820, A119821, A119815, A119817.
%K A119819 nonn
%O A119819 1,3
%A A119819 _Paul D. Hanna_, May 31 2006