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%I #8 Sep 20 2020 19:12:48
%S 1,1,6,57,683,9474,145815,2430393,43202448,810629805,15938815794,
%T 326653743510,6949638584208,153009877730525,3477623225388063,
%U 81429702521625843,1961136442605508341,48513571089988199157
%N Nonzero coefficients of the g.f. that satisfies: A(x) = x + A(A(x))^3.
%H Paul D. Hanna, <a href="/A153851/b153851.txt">Table of n, a(n) for n = 1..200</a>
%F G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) satisfies:
%F (1) A(x) = Series_Reversion( x - A(x)^3 ).
%F (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n) / n!. - _Paul D. Hanna_, Sep 07 2020
%F (3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/x / n! ). - _Paul D. Hanna_, Sep 07 2020
%F (4) x = A(A( x-x^3 - A(x)^3 )). - _Paul D. Hanna_, Sep 07 2020
%e G.f.: A(x) = x + x^3 + 6*x^5 + 57*x^7 + 683*x^9 + 9474*x^11 +...
%e A(x - A(x)^3) = x where
%e A(x)^3 = x^3 + 3*x^5 + 21*x^7 + 208*x^9 + 2517*x^11 + 34851*x^13 +...
%e SYSTEM OF RELATED FUNCTIONS.
%e A = A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations:
%e A = 1 + x^2*B^3;
%e B = A + x^2*C^3;
%e C = B + x^2*D^3;
%e D = C + x^2*E^3;
%e E = D + x^2*F^3; ...
%e where the functions xB, xC, xD, etc., are successive iterations of A(x):
%e x*A = A(x),
%e x*B = A(A(x)) = g.f. of A153852,
%e x*C = A(A(A(x))) = g.f. of A153853,
%e x*D = A(A(A(A(x)))) = g.f. of A153854, etc.
%e The nonzero coefficients of these functions begin:
%e A:[1, 1, 6, 57, 683, 9474, 145815, 2430393, 43202448,...];
%e B:[1, 2, 15, 165, 2213, 33693, 561867, 10053141, 190489374,...];
%e C:[1, 3, 27, 339, 5067, 84738, 1536867, 29687772, 603835479,...];
%e D:[1, 4, 42, 594, 9827, 179928, 3545637, 73988631, 1618178067,...];
%e E:[1, 5, 60, 945, 17180, 342765, 7316178, 164606166, 3866962617,...];
%e F:[1, 6, 81, 1407, 27918, 603879, 13907133, 336334443, 8466942393,...];
%e G:[1, 7, 105, 1995, 42938, 1001973, 24795645, 642380025, 17278647147,...];
%e H:[1, 8, 132, 2724, 63242, 1584768, 41975610, 1160887350, 33260962995,..]; ...
%e The main diagonal in the above table is A153850.
%o (PARI) {a(n)=local(A=x+x^2); for(i=0, n, A=serreverse(x-subst(A^3, x, x+x^2*O(x^(2*n))))) ; polcoeff(A, 2*n-1)}
%Y Cf. A153852, A153853, A153854, A153850; variants: A139702, A213591.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Jan 21 2009
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