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%I #17 May 06 2021 06:53:37
%S 1,1,3,7,15,32,65,131,260,501,965,1825,3419,6326,11652,21230,38405,
%T 69015,123334,218980,386809,679757,1189360,2071761,3594325,6211826,
%U 10698409,18363038,31420994,53605525,91198970,154746133,261929303,442310873,745264674,1253081340,2102754561
%N Expansion of Product_{k>=1} 1/(1 - x^k * (1 + k*x)).
%H Seiichi Manyama, <a href="/A336976/b336976.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (1 + k/d * x)^d / d).
%F a(n) ~ c * phi^(n+1) / sqrt(5), where c = Product_{k>=2} 1/(1 - 1/phi^k*(1 + k/phi)) = 167.5661037860673786430316975350024960626825333609486463342... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, May 06 2021
%t m = 36; CoefficientList[Series[Product[1/(1 - x^k*(1 + k*x)), {k, 1, m}], {x, 0, m}], x] (* _Amiram Eldar_, May 01 2021 *)
%o (PARI) N=66; x='x+O('x^N); Vec(1/prod(k=1, N, 1-x^k*(1+k*x)))
%o (PARI) N=66; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, (1+k/d*x)^d/d))))
%Y Cf. A227681, A336975, A336977, A336978, A336979, A336980.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Aug 09 2020