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A323683
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G.f.: Sum_{n>=0} x^n * ((1+x)^n + sqrt(3)*i)^n / (1 + sqrt(3)*i*x*(1+x)^n)^(n+1), where i^2 = -1.
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5
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1, 1, 2, 11, 34, 147, 741, 3723, 20622, 122611, 765147, 5039624, 34856671, 251799740, 1895620770, 14834210103, 120368044792, 1010721096231, 8766998042793, 78426199305392, 722487492991540, 6845217410165959, 66620876210526469, 665308346568565094, 6810637828466836635, 71400562836982319210, 765934679671944152100
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OFFSET
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0,3
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COMMENTS
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Note that the generating function expands into a power series in x with only real integer coefficients.
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LINKS
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FORMULA
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G.f.: Sum_{n>=0} x^n * ((1+x)^n + sqrt(3)*i)^n / (1 + sqrt(3)*i*x*(1+x)^n)^(n+1).
G.f.: Sum_{n>=0} x^n * ((1+x)^n - sqrt(3)*i)^n / (1 - sqrt(3)*i*x*(1+x)^n)^(n+1).
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 34*x^4 + 147*x^5 + 741*x^6 + 3723*x^7 + 20622*x^8 + 122611*x^9 + 765147*x^10 + 5039624*x^11 + 34856671*x^12 + ...
Let r = sqrt(3)*i, so that r^2 = -3, then
A(x) = 1/(1 + r*x) + x*((1+x) + r)/(1 + r*x*(1+x))^2 + x^2*((1+x)^2 + r)^2/(1 + r*x*(1+x)^2)^3 + x^3*((1+x)^3 + r)^3/(1 + r*x*(1+x)^3)^4 + x^4*((1+x)^4 + r)^4/(1 + r*x*(1+x)^4)^5 + x^5*((1+x)^5 + r)^5/(1 + r*x*(1+x)^5)^6 + ...
also,
A(x) = 1/(1 - r*x) + x*((1+x) - r)/(1 - r*x*(1+x))^2 + x^2*((1+x)^2 - r)^2/(1 - r*x*(1+x)^2)^3 + x^3*((1+x)^3 - r)^3/(1 - r*x*(1+x)^3)^4 + x^4*((1+x)^4 - r)^4/(1 - r*x*(1+x)^4)^5 + x^5*((1+x)^5 - r)^5/(1 - r*x*(1+x)^5)^6 + ...
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PROG
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(PARI) {a(n) = my(r = sqrt(3)*I, A = sum(m=0, n+1, x^m*((1+x +x*O(x^n))^m + r)^m/(1 + r*x*(1+x +x*O(x^n))^m)^(m+1) )); round(polcoeff(A, n))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(r = sqrt(3)*I, A = sum(m=0, n+1, x^m*((1+x +x*O(x^n))^m - r)^m/(1 - r*x*(1+x +x*O(x^n))^m)^(m+1) )); round(polcoeff(A, n))}
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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