|
%I #13 Dec 03 2022 12:04:33
%S 1,2,4,28,129,784,4547,28474,178947,1160189,7599423,50580502,
%T 339862004,2306662818,15774817084,108652754620,752854936635,
%U 5244889634762,36713446985136,258094902741010,1821402519619699,12898863644572142,91638273993427991,652926934710002885
%N a(n) = coefficient of x^n in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^n * A(x)^n.
%C Related identity: 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n)^(n-1), which holds when 0 < |x| < 1.
%C Related identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - x^n)^n, which holds when 0 < |x| < 1.
%H Paul D. Hanna, <a href="/A357158/b357158.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following conditions.
%F (1) 1 = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^n * A(x)^n.
%F (2) 1 = Sum_{n=-oo..+oo} (-1)^(n-1) * n * x^(n*(n-1)) / ((1 - x^n)^n * A(x)^n).
%e G.f.: A(x) = 1 + 2*x + 4*x^2 + 28*x^3 + 129*x^4 + 784*x^5 + 4547*x^6 + 28474*x^7 + 178947*x^8 + 1160189*x^9 + 7599423*x^10 + 50580502*x^11 + 339862004*x^12 + ...
%e where
%e 1 = ... - 3*(x*A(x))^(-3)/(1 - x^(-3))^4 - 2*(x*A(x))^(-2)/(1 - x^(-2))^3 - (x*A(x))^(-1)/(1 - x^(-1))^2 + 0 + x*A(x) + 2*(x*A(x))^2*(1 - x^2) + 3*(x*A(x))^3*(1 - x^3)^2 + 4*(x*A(x))^4*(1 - x^4)^3 + ... + n*(x*A(x))^n*(1 - x^n)^(n-1) + ...
%e SPECIFIC VALUES.
%e A(1/8) = 1.526583049137946431...
%e A(1/9) = 1.373618404532597561...
%e A(1/10) = 1.3014404438931094722929945...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = polcoeff( sum(n=-#A,#A, n * x^n * (1 - x^n +x*O(x^#A) )^n * Ser(A)^n ),#A-1) );A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%K nonn
%O 0,2
%A _Paul D. Hanna_, Oct 05 2022
| 