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A165941
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G.f.: A(x) = exp( Sum_{n>=1} 2^n * x^n/(n*(1+x^n)) ).
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6
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1, 2, 2, 6, 10, 18, 42, 78, 154, 314, 626, 1246, 2498, 4994, 9970, 19974, 39930, 79826, 159706, 319374, 638714, 1277530, 2554978, 5109854, 10219922, 20439714, 40879234, 81758854, 163517466, 327034514, 654069866, 1308139246, 2616277578
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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 + 2/(1+x - 2*x/(1+x^2 - 2*x^2/(1+x^3 - 2*x^3/(1+x^4 - 2*x^4/(1+x^5 - 2*x^5/(1+x^6 - 2*x^6/(1+x^7 - 2*x^7/(1+x^8 - 2*x^8/(...))))))))), a continued fraction.
G.f.: A(x) = (1 + x*B(x))/(1 - x*B(x)), where B(x) = (1 + x^2*C(x))/(1 - x^2*C(x)), C(x) = (1 + x^3*D(x))/(1 - x^3*D(x)), D(x) = (1 + x^4*E(x))/(1 - x^4*E(x)), ... - Paul D. Hanna, Jun 14 2015
a(n) ~ c * 2^n, where c = 2^(7/8) / EllipticTheta(2, 0, 1/sqrt(2)) = 0.6091497110662286155211146043057245512950999410185846... - Vaclav Kotesovec, Oct 18 2020, updated Apr 18 2024
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 2*x^2 + 6*x^3 + 10*x^4 + 18*x^5 + 42*x^6 + 78*x^7 +...
such that
log(A(x)) = 2*x/(1+x) + 2^2*x^2/(2*(1+x^2)) + 2^3*x^3/(3*(1+x^3)) + 2^4*x^4/(4*(1+x^4)) + 2^5*x^5/(5*(1+x^5)) +...
Also, A(x) = (1 + x*B(x))/(1 - x*B(x)), where
B(x) = 1 + 2*x^2 + 2*x^4 + 4*x^5 + 2*x^6 + 8*x^7 + 6*x^8 + 20*x^9 + 18*x^10 + 36*x^11 + 54*x^12 + 76*x^13 + 150*x^14 + 172*x^15 +...
such that B(x) = (1 + x*C(x))/(1 - x*C(x)), where
C(x) = 1 + 2*x^3 + 2*x^6 + 4*x^7 + 2*x^9 + 8*x^10 + 4*x^11 + 10*x^12 + 12*x^13 + 16*x^14 + 22*x^15 + 32*x^16 + 44*x^17 + 66*x^18 +...
such that C(x) = (1 + x*D(x))/(1 - x*D(x)), where
D(x) = 1 + 2*x^4 + 2*x^8 + 4*x^9 + 2*x^12 + 8*x^13 + 4*x^14 + 8*x^15 + 2*x^16 + 12*x^17 + 16*x^18 + 20*x^19 + 18*x^20 + 24*x^21 +...
such that D(x) = (1 + x*E(x))/(1 - x*E(x)), where
E(x) = 1 + 2*x^5 + 2*x^10 + 4*x^11 + 2*x^15 + 8*x^16 + 4*x^17 + 8*x^18 + 2*x^20 + 12*x^21 + 16*x^22 + 20*x^23 + 16*x^24 + 10*x^25 +...
such that E(x) = (1 + x*F(x))/(1 - x*F(x)), where
F(x) = 1 + 2*x^6 + 2*x^12 + 4*x^13 + 2*x^18 + 8*x^19 + 4*x^20 + 8*x^21 + 2*x^24 + 12*x^25 + 16*x^26 + 20*x^27 + 16*x^28 + 8*x^29 + 18*x^30 + 16*x^31 + 36*x^32 +...
etc.
The coefficients in the above functions tend toward the terms in triangle A259192.
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MATHEMATICA
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nmax = 40; CoefficientList[Series[Exp[Sum[2^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, 2^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 - 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,changed
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AUTHOR
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STATUS
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approved
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