A392205 - OEIS

%I #11 Jan 06 2026 10:05:47

%S 1,1,3,7,25,78,303,1104,4536,18174,78042,333204,1484446,6635245,

%T 30501324,141223812,666998868,3177455112,15366908520,74979351720,

%U 370318109661,1844809350815,9285027779942,47116445985240,241245182698771,1244795754653734,6474962893493622,33926096501154962

%N G.f. A(x) satisfies A(x - A(x)) = x^3 + x^4.

%H Paul D. Hanna, <a href="/A392205/b392205.txt">Table of n, a(n) for n = 3..403</a>

%F G.f. A(x) = Sum_{n>=3} a(n)*x^n satisfies the following formulas.

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

%F (2) A(x - x^3 - x^4 - A(x)) = (x - A(x))^3 + (x - A(x))^4.

%F (3) A(x) = B(x)^3 + B(x)^4 where B(x - A(x)) = x and B(x) = x + A(B(x)).

%F (4) B(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n / n! where B(x - A(x)) = x.

%F (5) B(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (1/x)*A(x)^n / n! ) where B(x - A(x)) = x.

%e A(x) = x^3 + x^4 + 3*x^5 + 7*x^6 + 25*x^7 + 78*x^8 + 303*x^9 + 1104*x^10 + 4536*x^11 + 18174*x^12 + 78042*x^13 + 333204*x^14 + ...

%e where A(x - A(x)) = x^3 + x^4.

%e RELATED SERIES.

%e Let B(x) satisfy B(x - A(x)) = x, then

%e B(x) = x + x^3 + x^4 + 6*x^5 + 14*x^6 + 65*x^7 + 213*x^8 + 943*x^9 + 3634*x^10 + 16191*x^11 + 68263*x^12 + ...

%e where B(x) = x + B(B(x))^3 + B(B(x))^4 and B(x) = x + A(B(x)).

%e Also, from B(x) given above, we have the related series

%e B(x)^3 = x^3 + 3*x^5 + 3*x^6 + 21*x^7 + 48*x^8 + 235*x^9 + 762*x^10 + 3432*x^11 + 13153*x^12 + ...

%e B(x)^4 = x^4 + 4*x^6 + 4*x^7 + 30*x^8 + 68*x^9 + 342*x^10 + 1104*x^11 + 5021*x^12 + ...

%e where A(x) = B(x)^3 + B(x)^4.

%o (PARI) \\ Using A(x) = B(x)^3 + B(x)^4 where B(x - A(x)) = x.

%o {a(n) = my(A = x^3); for(i=1, n, B = serreverse(x - A + x^3*O(x^n));

%o A = B^3 + B^4); polcoef(A, n)}

%o for(n=3, 30, print1(a(n), ", "))

%Y Cf. A392204.

%K nonn

%O 3,3

%A _Paul D. Hanna_, Jan 05 2026