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Expansion of g.f. A(x) satisfying A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9).
2

%I #17 Aug 16 2023 10:26:23

%S 1,1,3,18,126,966,7863,66696,583111,5217513,47547405,439777242,

%T 4117802109,38956162023,371795456373,3575401032544,34611064585803,

%U 336998629754631,3298200003722997,32428037256038775,320151289224740949,3172536384239678856,31544584654878015766

%N Expansion of g.f. A(x) satisfying A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9).

%C Compare to: G(x)^3 = 1 + x*(G(x) + G(x)^2 + G(x)^3) holds when G(x) = 1/(1-x).

%C Conjecture: a(n) == 0 (mod 3) for n > 0 except when n == 1 (mod 7).

%H Paul D. Hanna, <a href="/A363560/b363560.txt">Table of n, a(n) for n = 0..500</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.

%F (1) A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^4 - A(x)^6 + A(x)^7).

%F (2) A(x)^2 = 1 + x*(A(x) + A(x)^2 - A(x)^3 + A(x)^5 - A(x)^6 + A(x)^8) .

%F (3) A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9).

%F (4) A(x)^4 = 1 + x*(A(x) + A(x)^2 + A(x)^4 - A(x)^6 + A(x)^7 + A(x)^10).

%F (5) A(x)^5 = 1 + x*(A(x) + A(x)^2 + A(x)^4 + A(x)^5 - A(x)^6 + A(x)^8 + A(x)^11).

%F (6) A(x)^6 = 1 + x*(A(x) + A(x)^2 + A(x)^4 + A(x)^5 + A(x)^9 + A(x)^12).

%F (7) A(x) = (1/x) * Series_Reversion( x/(1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^5) ) ) ).

%e G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 126*x^4 + 966*x^5 + 7863*x^6 + 66696*x^7 + 583111*x^8 + 5217513*x^9 + 47547405*x^10 + ...

%e such that

%e A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9).

%e Also,

%e A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^4 - A(x)^6 + A(x)^7).

%e RELATED TABLE.

%e The table of coefficients in A(x)^n begins:

%e n=1: [1, 1, 3, 18, 126, 966, 7863, 66696, ...];

%e n=2: [1, 2, 7, 42, 297, 2292, 18738, 159450, ...];

%e n=3: [1, 3, 12, 73, 522, 4059, 33354, 284886, ...];

%e n=4: [1, 4, 18, 112, 811, 6360, 52566, 450888, ...];

%e n=5: [1, 5, 25, 160, 1175, 9301, 77370, 666780, ...];

%e n=6: [1, 6, 33, 218, 1626, 13002, 108919, 943524, ...];

%e n=7: [1, 7, 42, 287, 2177, 17598, 148540, 1293937, ...];

%e n=8: [1, 8, 52, 368, 2842, 23240, 197752, 1732928, ...];

%e n=9: [1, 9, 63, 462, 3636, 30096, 258285, 2277756, ...];

%e ...

%e from which one can verify the formulas involving powers of A(x).

%e RELATED SERIES.

%e Let G(x) = 1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^5) )

%e where

%e G(x) = 1 + x + 2*x^2 + 11*x^3 + 61*x^4 + 380*x^5 + 2502*x^6 + 17163*x^7 + 121312*x^8 + 877370*x^9 + 6461765*x^10 + ...

%e then

%e A(x) = G(x*A(x)),

%e and so

%e A(x) = (1/x) * Series_Reversion( x/G(x) );

%e thus,

%e x*A(x) = (A(x) - 1) / (1 + (A(x) - 1)*(A(x)^2 + A(x)^5) )

%e which is equivalent to

%e A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^4 - A(x)^6 + A(x)^7).

%e TERMS MODULO 3.

%e It appears that a(n) == 0 (mod 3) for n > 0 except when n == 1 (mod 7).

%e The residues of a(7*k + 1) modulo 3, for k >= 0, begin

%e a(7*k + 1) (mod 3) = [1, 1, 1, 1, 0, 2, 1, 0, 0, 1, 1, 2, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 1, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, ...].

%o (PARI) {a(n) = my(A=1+x); for(i=1, n, A = (1 + x*(A + A^2 + A^9) +x*O(x^n))^(1/3) ); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A300048, A161634.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 12 2023