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A326604 G.f. A(x) satisfies x / Series_Reversion(x*A(x)) = (2*A(x) + 1+x)/3.

%I A326604 #15 Nov 03 2019 06:46:56
%S A326604 1,1,3,21,231,3333,58167,1175877,26827623,679078677,18844334727,
%T A326604 568229240901,18491559492999,645850960844469,24099045218945031,
%U A326604 956889503377128261,40291822946545245927,1793614919867776690389,84177429562216608349959,4154548653801498090246597,215137302566817565660007367,11664210072689092804458508533
%N A326604 G.f. A(x) satisfies x / Series_Reversion(x*A(x)) = (2*A(x) + 1+x)/3.
%C A326604 a(k) = 6 (mod 9) when k = A191107(n) for n > 1 (conjecture).
%C A326604 a(n) = 3*A249933(n) for n > 1.
%H A326604 Paul D. Hanna, <a href="/A326604/b326604.txt">Table of n, a(n) for n = 0..400</a>
%F A326604 G.f. A(x) satisfies:
%F A326604 (1) A( 3*x/(2*A(x) + 1+x) ) = (2*A(x) + 1+x)/3.
%F A326604 (2) A(x) = (1 + 2*A(x*A(x))) / (3-x).
%F A326604 (3) A(x) = 1 + Sum_{n>=1} G^n(x) * 2^(n-1)/3^n where G(x) = x*A(x) and G^n(x) = G^{n-1}(x*A(x)) denotes iteration with G^0(x) = x.
%e A326604 G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 231*x^4 + 3333*x^5 + 58167*x^6 + 1175877*x^7 + 26827623*x^8 + 679078677*x^9 + 18844334727*x^10 +...
%e A326604 such that
%e A326604 x/Series_Reversion(x*A(x)) = (2*A(x) + 1+x)/3 = 1 + x + 2*x^2 + 14*x^3 + 154*x^4 + 2222*x^5 + 38778*x^6 + 783918*x^7 + 17885082*x^8 + 452719118*x^9 + ...
%e A326604 ITERATIONS OF x*A(x).
%e A326604 Let G(x) = x*A(x), then
%e A326604 A(x) = 1 + G(x)/3 + G(G(x))*2/3^2 + G(G(G(x)))*2^2/3^3 + G(G(G(G(x))))*2^3/3^4 + G(G(G(G(G(x)))))*2^4/3^5 +...
%e A326604 The table of coefficients in the iterations of x*A(x) begin:
%e A326604 [1, 1, 3, 21, 231, 3333, 58167, 1175877, 26827623, ...];
%e A326604 [1, 2, 8, 58, 630, 8958, 154530, 3096330, 70161318, ...];
%e A326604 [1, 3, 15, 117, 1285, 18167, 310735, 6177745, 139076385, ...];
%e A326604 [1, 4, 24, 204, 2308, 32800, 559124, 11053668, 247451528, ...];
%e A326604 [1, 5, 35, 325, 3835, 55365, 946623, 18671961, 416326935, ...];
%e A326604 [1, 6, 48, 486, 6026, 89158, 1539350, 30423134, 677231222, ...];
%e A326604 [1, 7, 63, 693, 9065, 138383, 2427943, 48304893, 1076756889, ...];
%e A326604 [1, 8, 80, 952, 13160, 208272, 3733608, 75127944, 1682704256, ...];
%e A326604 [1, 9, 99, 1269, 18543, 305205, 5614887, 114768093, 2592154167, ...]; ...
%e A326604 in which the following sum along column k equals a(k+1):
%e A326604 a(2) = 3 = 1/3 + 2*2/9 + 3*4/27 + 4*8/81 + 5*16/243 + 6*32/729 +...
%e A326604 a(3) = 21 = 3/3 + 8*2/9 + 15*4/27 + 24*8/81 + 35*16/243 + 48*32/729 + ...
%e A326604 a(4) = 231 = 21/3 + 58*2/9 + 117*4/27 + 204*8/81 + 325*16/243 + 486*32/729 +...
%e A326604 a(5) = 3333 = 231*2/3 + 630*2/9 + 1285*4/27 + 2308*8/81 + 3835*16/243 + 6026*32/729 +...
%t A326604 nmax = 21; sol = {a[0] -> 1}; Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + 2 A[x A[x] + O[x]^(n+1)])/(3-x), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
%t A326604 sol /. Rule -> Set;
%t A326604 a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 03 2019 *)
%o A326604 (PARI) /* Prints N terms using x/Series_Reversion(x*A(x)) = (2*A(x) + 1+x)/3 */
%o A326604 N = 30; {A=[1, 1]; for(i=1, N, A = concat(A, -3*Vec(x/serreverse(x*Ser(concat(A, 0))))[#A+1]); print1(i, ", ") ); A}
%Y A326604 Cf. A120956, A249933.
%K A326604 nonn
%O A326604 0,3
%A A326604 _Paul D. Hanna_, Sep 21 2019