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A357799 a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * (A(x) + x^n)^(n+1). 1

%I #9 Jan 01 2023 05:42:35

%S 1,1,4,10,33,105,363,1268,4600,16954,63663,242180,932255,3623239,

%T 14200924,56061965,222728379,889828825,3572675122,14408128581,

%U 58338540673,237067134533,966522205819,3952323714926,16206324436147,66621153183615,274505283101713

%N a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * (A(x) + x^n)^(n+1).

%H Paul D. Hanna, <a href="/A357799/b357799.txt">Table of n, a(n) for n = 0..400</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.

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

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

%F a(n) ~ c * d^n / n^(3/2), where d = 4.3661068223321816847380533247926... and c = 0.8485754894190316922838386890774... - _Vaclav Kotesovec_, Jan 01 2023

%e G.f.: A(x) = 1 + x + 4*x^2 + 10*x^3 + 33*x^4 + 105*x^5 + 363*x^6 + 1268*x^7 + 4600*x^8 + 16954*x^9 + 63663*x^10 + 242180*x^11 + 932255*x^12 + ...

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

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

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 23 2022

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Last modified July 19 11:36 EDT 2024. Contains 374394 sequences. (Running on oeis4.)