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A357158
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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.
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5
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1, 2, 4, 28, 129, 784, 4547, 28474, 178947, 1160189, 7599423, 50580502, 339862004, 2306662818, 15774817084, 108652754620, 752854936635, 5244889634762, 36713446985136, 258094902741010, 1821402519619699, 12898863644572142, 91638273993427991, 652926934710002885
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OFFSET
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0,2
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COMMENTS
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Related identity: 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n)^(n-1), which holds when 0 < |x| < 1.
Related identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - x^n)^n, which holds when 0 < |x| < 1.
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following conditions.
(1) 1 = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^n * A(x)^n.
(2) 1 = Sum_{n=-oo..+oo} (-1)^(n-1) * n * x^(n*(n-1)) / ((1 - x^n)^n * A(x)^n).
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EXAMPLE
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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 + ...
where
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) + ...
SPECIFIC VALUES.
A(1/8) = 1.526583049137946431...
A(1/9) = 1.373618404532597561...
A(1/10) = 1.3014404438931094722929945...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
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]}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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