A370032 - OEIS

%I #10 Feb 12 2024 02:26:54

%S 1,-1,0,3,-10,21,-25,-23,228,-737,1479,-1245,-4352,25206,-72761,

%T 128245,-38615,-697798,3109043,-8016819,11763729,6510069,-108216128,

%U 403917707,-925174519,1025709534,2228869018,-16585014721,53758505915,-107811969706,69758146717,478423936550,-2520835801152,7208714823250

%N Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n + A(x))^n = 1 + 3*Sum_{n>=1} x^(n^2).

%H Paul D. Hanna, <a href="/A370032/b370032.txt">Table of n, a(n) for n = 1..401</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

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

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

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

%F (4) Sum_{n=-oo..+oo} x^(n^2) / (1 + x^n*A(x))^n = 1 + 3*Sum_{n>=1} x^(n^2).

%F (5) Sum_{n=-oo..+oo} x^(n^2) / (1 - x^n*A(x))^(n+1) = 1 + 3*Sum_{n>=1} x^(n^2).

%F (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + x^n*A(x))^n = 0.

%e G.f.: A(x) = x - x^2 + 3*x^4 - 10*x^5 + 21*x^6 - 25*x^7 - 23*x^8 + 228*x^9 - 737*x^10 + 1479*x^11 - 1245*x^12 - 4352*x^13 + 25206*x^14 - 72761*x^15 + 128245*x^16 + ...

%e where

%e Sum_{n=-oo..+oo} (x^n + A(x))^n = 1 + 3*x + 3*x^4 + 3*x^9 + 3*x^16 + 3*x^25 + 3*x^36 + 3*x^49 + 3*x^64 + ...

%o (PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);A[#A] = -polcoeff( sum(m=-#A, #A, (x^m + Ser(A))^m ) - 1 - 3*sum(m=1, #A, x^(m^2) ), #A-1) ); A[n+1]}

%o for(n=1, 30, print1(a(n), ", "))

%Y Cf. A370041, A370030, A370031, A355868, A370033, A370034, A370035, A370036, A370037, A370038, A370039, A370043.

%K sign

%O 1,4

%A _Paul D. Hanna_, Feb 11 2024