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A153850 a(n) is the coefficient of x^(2*n-1) in the n-fold self-composition of G(x) = x + G(G(x))^3 = g.f. of A153851.

%I A153850 #4 Mar 10 2023 07:36:28
%S A153850 1,2,27,594,17180,603879,24795645,1160887350,60940292571,
%T A153850 3541938123306,225669592036086,15634133444509443,1169781625911185118,
%U A153850 93989088711427170141,8069678384570571946581,737204558292074214218778
%N A153850 a(n) is the coefficient of x^(2*n-1) in the n-fold self-composition of G(x) = x + G(G(x))^3 = g.f. of A153851.
%e A153850 Let A(x) be the g.f. of A153851, which begins
%e A153850 A(x) = x + x^3 + 6*x^5 + 57*x^7 + 683*x^9 + 9474*x^11 + 145815*x^13 + 2430393*x^15 + 43202448*x^17 + ... + A153851(2*n-1)*x^n + ...
%e A153850 then A(x) satisfies A(x - A(x)^3) = x.
%e A153850 Further, let successive iterations of A(x) be denoted by
%e A153850 B(x) = A(A(x)) = g.f. of A153852,
%e A153850 C(x) = A(A(A(x))) = g.f. of A153853,
%e A153850 D(x) = A(A(A(A(x)))) = g.f. of A153854, etc.,
%e A153850 then the nonzero coefficients in the successive iterations of A(x) form the table:
%e A153850 A:[1, 1, 6, 57, 683, 9474, 145815, 2430393, ...];
%e A153850 B:[1, 2, 15, 165, 2213, 33693, 561867, 10053141, ...];
%e A153850 C:[1, 3, 27, 339, 5067, 84738, 1536867, 29687772, ...];
%e A153850 D:[1, 4, 42, 594, 9827, 179928, 3545637, 73988631, ...];
%e A153850 E:[1, 5, 60, 945, 17180, 342765, 7316178, 164606166, ...];
%e A153850 F:[1, 6, 81, 1407, 27918, 603879, 13907133, 336334443, ...];
%e A153850 G:[1, 7, 105, 1995, 42938, 1001973, 24795645, 642380025, ...];
%e A153850 H:[1, 8, 132, 2724, 63242, 1584768, 41975610, 1160887350, ...]; ...
%e A153850 in which the main diagonal equals this sequence.
%o A153850 (PARI) {a(n) = my(G=x+O(x^(2*n+1)),H=G); for(i=0, n, G=serreverse(x-G^3)); for(i=1,n,H=subst(G,x,H)); polcoeff(H, 2*n-1)}
%o A153850 for(n=1,20,print1(a(n),", "))
%Y A153850 Cf. A153851, A153852, A153853, A153854.
%K A153850 nonn
%O A153850 1,2
%A A153850 _Paul D. Hanna_, Jan 21 2009