A366733 - OEIS

%I #5 Oct 29 2023 22:02:33

%S 1,3,12,90,702,5838,50895,458103,4225683,39745665,379730658,

%T 3674980518,35951809104,354950991006,3532167377340,35390917028619,

%U 356742401734236,3615164398809324,36809446799831823,376387507560832992,3863438843523528636,39794189982905311407

%N Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (3 - x^(n-1))^(n+1).

%C a(n) = Sum_{k=0..n} A366730(n,k) * 3^k for n >= 0.

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

%F (1) 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (3 - x^(n-1))^(n+1).

%F (2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( A(x)^n * (1 - 3*x^(n+1))^(n-1) ).

%e G.f.: A(x) = 1 + 3*x + 12*x^2 + 90*x^3 + 702*x^4 + 5838*x^5 + 50895*x^6 + 458103*x^7 + 4225683*x^8 + 39745665*x^9 + 379730658*x^10 + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoeff( sum(n=-#A,#A, x^n * Ser(A)^n * (3 - x^(n-1))^(n+1) ), #A-2));A[n+1]}

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

%Y Cf. A366730, A366731, A366732, A366734, A366735.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 29 2023