A358937 - OEIS

%I #33 Oct 30 2023 12:05:10

%S 1,1,3,6,13,31,76,192,504,1351,3668,10082,27991,78335,220778,626141,

%T 1785593,5117179,14729826,42568767,123465517,359268141,1048541699,

%U 3068583485,9002849260,26474484680,78019959584,230381635121,681544367457,2019718168994,5995000501189

%N Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (x^(2*n) - A(x))^n.

%C Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(2*n+1))^n, which holds formally for all y.

%H Paul D. Hanna, <a href="/A358937/b358937.txt">Table of n, a(n) for n = 0..200</a>

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

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

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

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

%F (4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 - x^(2*n)*A(x))^n ) (trivial).

%e G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 13*x^4 + 31*x^5 + 76*x^6 + 192*x^7 + 504*x^8 + 1351*x^9 + 3668*x^10 + 10082*x^11 + 27991*x^12 + ...

%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 * (x^(2*n) - Ser(A))^n ), #A) );A[n+1]}

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

%Y Cf. A366229, A355865.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 07 2022