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A227681 G.f.: exp( Sum_{n>=1} x^n * (1+x)^n / (n*(1-x^n)) ).

%I A227681 #20 Jul 31 2021 09:22:39
%S A227681 1,1,3,6,12,23,43,79,142,252,442,766,1316,2244,3799,6393,10704,17841,
%T A227681 29618,49000,80823,132964,218242,357501,584608,954553,1556575,2535425,
%U A227681 4125805,6708143,10898897,17696749,28719276,46586050,75538702,122444483,198420445,321461918
%N A227681 G.f.: exp( Sum_{n>=1} x^n * (1+x)^n / (n*(1-x^n)) ).
%H A227681 Seiichi Manyama, <a href="/A227681/b227681.txt">Table of n, a(n) for n = 0..1000</a>
%F A227681 G.f.: exp( Sum_{n>=1} x^n * Sum_{d|n} (1+x)^d / d ).
%F A227681 G.f.: Product {n >= 1} 1/(1 - (1 + x)*x^n). - _Peter Bala_, Jan 20 2015
%F A227681 a(n) ~ phi^(n+1) / (sqrt(5)* A276987), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jul 16 2019
%e A227681 G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 12*x^4 + 23*x^5 + 43*x^6 + 79*x^7 +...
%e A227681 where
%e A227681 log(A(x)) = x*(1+x)/(1-x) + x^2*(1+x)^2/(2*(1-x^2)) + x^3*(1+x)^3/(3*(1-x^3)) + x^4*(1+x)^4/(4*(1-x^4)) + x^5*(1+x)^5/(5*(1-x^5)) +...
%e A227681 Explicitly,
%e A227681 log(A(x)) = x + 5*x^2/2 + 10*x^3/3 + 17*x^4/4 + 26*x^5/5 + 38*x^6/6 + 57*x^7/7 + 81*x^8/8 + 118*x^9/9 + 180*x^10/10 +...
%t A227681 nmax = 50; CoefficientList[Series[Product[1/(1 - x^k*(1 + x)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 31 2021 *)
%o A227681 (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*(1+x)^m/(1-x^m +x*O(x^n)) )), n)}
%o A227681 for(n=0, 50, print1(a(n), ", "))
%o A227681 (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m*sumdiv(m, d, (1+x +x*O(x^n))^d/d) )), n)}
%o A227681 for(n=0, 50, print1(a(n), ", "))
%Y A227681 Cf. A160571, A227682, A306565, A306749.
%K A227681 nonn
%O A227681 0,3
%A A227681 _Paul D. Hanna_, Jul 19 2013