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A259276
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G.f.: A(x) = exp( Sum_{n>=1} 6^n * x^n/(n*(1+x^n)) ).
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4
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1, 6, 30, 186, 1110, 6630, 39846, 239010, 1433910, 8603790, 51622446, 309733890, 1858404990, 11150428470, 66902565630, 401415404586, 2408492418870, 14450954480790, 86705726950470, 520234361647890, 3121406169699270, 18728437018590366, 112370622111206670, 674223732666113010
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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).
In general, for m > 1, if g.f. = exp(Sum_{k>=1} m^k * x^k/(k*(1+x^k))) then a(n) ~ 2 * m^(n - 1/8) / EllipticTheta(2, 0, 1/sqrt(m)). - Vaclav Kotesovec, Apr 18 2024
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LINKS
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FORMULA
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G.f.: -1/5 + (6/5)/(1+x - 6*x/(1+x^2 - 6*x^2/(1+x^3 - 6*x^3/(1+x^4 - 6*x^4/(1+x^5 - 6*x^5/(1+x^6 - 6*x^6/(1+x^7 - 6*x^7/(1+x^8 - 6*x^8/(...))))))))), a continued fraction.
G.f.: A(x) = (1 + x*B(x))/(1 - 5*x*B(x)), where B(x) = (1 + x^2*C(x))/(1 - 5*x^2*C(x)), C(x) = (1 + x^3*D(x))/(1 - 5*x^3*D(x)), D(x) = (1 + x^4*E(x))/(1 - 5*x^4*E(x)), ...
a(n) ~ c * 6^n, where c = 2^(7/8) / (3^(1/8) * EllipticTheta(2, 0, 1/sqrt(6))) = 0.8537393061385536395511385858124987202486847622399194... - Vaclav Kotesovec, Oct 18 2020, updated Apr 18 2024
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EXAMPLE
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G.f.: A(x) = 1 + 6*x + 30*x^2 + 186*x^3 + 1110*x^4 + 6630*x^5 +...
such that
log(A(x)) = 6*x/(1+x) + 6^2*x^2/(2*(1+x^2)) + 6^3*x^3/(3*(1+x^3)) + 6^4*x^4/(4*(1+x^4)) + 6^5*x^5/(5*(1+x^5)) +...
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MATHEMATICA
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nmax = 30; CoefficientList[Series[Exp[Sum[6^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, 6^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 - 5*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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