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A323685
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G.f.: Sum_{n>=0} x^n * ((1+x)^n + sqrt(5)*i)^n / (1 + sqrt(5)*i*x*(1+x)^n)^(n+1), where i^2 = -1.
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5
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1, 1, 2, 15, 46, 207, 1201, 6283, 36746, 235463, 1553311, 10803272, 79101355, 602629168, 4775430042, 39306129479, 334963829368, 2949993280119, 26808687950425, 250987986961396, 2417350292179932, 23922186855590303, 242961589181754713, 2529832992050854458, 26980268905373556691, 294452742973863998098, 3285813227185410286520
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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(5)*i)^n / (1 + sqrt(5)*i*x*(1+x)^n)^(n+1).
G.f.: Sum_{n>=0} x^n * ((1+x)^n - sqrt(5)*i)^n / (1 - sqrt(5)*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 + 15*x^3 + 46*x^4 + 207*x^5 + 1201*x^6 + 6283*x^7 + 36746*x^8 + 235463*x^9 + 1553311*x^10 + 10803272*x^11 + ...
Let r = sqrt(5)*i, so that r^2 = -5, 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(5)*I, A = sum(m=0, n+2, 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(5)*I, A = sum(m=0, n+2, 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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