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

%I #8 Jan 22 2024 00:05:03

%S 1,1,1,1,3,3,8,12,29,48,105,202,420,831,1729,3538,7370,15293,32094,

%T 67410,142221,301074,639076,1360991,2903607,6213695,13318015,28616357,

%U 61576994,132779990,286704638,620144700,1343082108,2913091456,6325803831,13754042495,29937461161

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

%H Paul D. Hanna, <a href="/A369084/b369084.txt">Table of n, a(n) for n = 1..600</a>

%F G.f. A(x) satisfies:

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

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

%e G.f.: A(x) = x + x^2 + x^3 + x^4 + 3*x^5 + 3*x^6 + 8*x^7 + 12*x^8 + 29*x^9 + 48*x^10 + 105*x^11 + 202*x^12 + 420*x^13 + 831*x^14 + 1729*x^15 + ...

%e where

%e 1 = ... + x^6/(1/x^3 - A(x))^3 + x^3/(1/x^2 - A(x))^2 + x/(1/x - A(x)) + 1 + (x - A(x)) + x*(x^2 - A(x))^2 + x^3*(x^3 - A(x))^3 + x^6*(x^4 - A(x))^4 + ... + x^(n*(n-1)/2)*(x^n - A(x))^n + ...

%o (PARI) {a(n) = my(A=[0], M); for(i=1, n, A=concat(A, 0); M = ceil(sqrt(2*(#A)+9));

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

%o for(n=1, 40, print1(a(n), ", "))

%Y Cf. A355861.

%K nonn

%O 1,5

%A _Paul D. Hanna_, Jan 21 2024