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A259275
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G.f.: A(x) = exp( Sum_{n>=1} 5^n * x^n/(n*(1+x^n)) ).
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4
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1, 5, 20, 105, 520, 2580, 12945, 64680, 323320, 1616780, 8083745, 40418380, 202092620, 1010462480, 5052310420, 25261556205, 126307777920, 631538879180, 3157694416720, 15788472066780, 78942360284720, 394711801527505, 1973559007551520, 9867795037511480, 49338975188073020
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OFFSET
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0,2
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COMMENTS
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Compare to: exp( Sum_{n>=1} x^n/(1+x^n)/n ) = Sum_{n>=0} x^(n*(n+1)/2).
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LINKS
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FORMULA
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G.f.: -1/4 + (5/4)/(1+x - 5*x/(1+x^2 - 5*x^2/(1+x^3 - 5*x^3/(1+x^4 - 5*x^4/(1+x^5 - 5*x^5/(1+x^6 - 5*x^6/(1+x^7 - 5*x^7/(1+x^8 - 5*x^8/(...))))))))), a continued fraction.
G.f.: A(x) = (1 + x*B(x))/(1 - 4*x*B(x)), where B(x) = (1 + x^2*C(x))/(1 - 4*x^2*C(x)), C(x) = (1 + x^3*D(x))/(1 - 4*x^3*D(x)), D(x) = (1 + x^4*E(x))/(1 - 4*x^4*E(x)), ...
a(n) ~ c * 5^n, where c = 2 / (5^(1/8) * EllipticTheta(2, 0, 1/sqrt(5))) = 0.8277706439469762656495798472679610454060848013727259... - Vaclav Kotesovec, Oct 18 2020, updated Apr 18 2024
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EXAMPLE
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G.f.: A(x) = 1 + 5*x + 20*x^2 + 105*x^3 + 520*x^4 + 2580*x^5 +...
such that
log(A(x)) = 5*x/(1+x) + 5^2*x^2/(2*(1+x^2)) + 5^3*x^3/(3*(1+x^3)) + 5^4*x^4/(4*(1+x^4)) + 5^5*x^5/(5*(1+x^5)) +...
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MATHEMATICA
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nmax = 30; CoefficientList[Series[Exp[Sum[5^k * x^k / (1 + x^k)/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 18 2020 *)
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PROG
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(PARI) {a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, 5^m*x^m/(1+x^m+x*O(x^n))/m)), n))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1 + x^(n+1-i)*A)/(1 - 4*x^(n+1-i)*A+ x*O(x^n))); 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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