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A322209 L.g.f.: log( Product_{n>=1} 1/(1 - (2^n+1)*x^n) ).

%I A322209 #3 Dec 01 2018 12:03:32
%S A322209 0,3,19,54,199,408,1612,3090,11023,26487,80994,199686,676540,1700832,
%T A322209 5285096,15197274,45739039,131368404,401655943,1172222958,3549402474,
%U A322209 10533769146,31617172980,94336116834,283990486780,848323147233,2546924693306,7631598676410,22903854049016,68645946621360,206035134959112,617739968277066,1853594327953471
%N A322209 L.g.f.: log( Product_{n>=1} 1/(1 - (2^n+1)*x^n) ).
%F A322209 a(n) = Sum_{k=0..n} A322200(n-k,k) * 2^k for n >= 0.
%e A322209 L.g.f.: L(x) = 3*x + 19*x^2/2 + 54*x^3/3 + 199*x^4/4 + 408*x^5/5 + 1612*x^6/6 + 3090*x^7/7 + 11023*x^8/8 + 26487*x^9/9 + 80994*x^10/10 + 199686*x^11/11 + 676540*x^12/12 + ...
%e A322209 such that
%e A322209 exp( L(x) ) = 1 + 3*x + 14*x^2 + 51*x^3 + 195*x^4 + 663*x^5 + 2345*x^6 + 7707*x^7 + 25744*x^8 + 82980*x^9 + 267812*x^10 + 846150*x^11 + 2676163*x^12 + ... + A322199(n)*x^n + ...
%e A322209 also,
%e A322209 exp( L(x) ) = 1/( (1 - 3*x) * (1 - 5*x^2) * (1 - 9*x^3) * (1 - 17*x^4) * (1 - 33*x^5) * (1 - 65*x^6) * (1 - 129*x^7) * ... * (1 - (2^n+1)*x^n) * ... ).
%o A322209 (PARI)
%o A322209 {L = sum(n=1,41, -log(1 - (x^n + y^n) +O(x^41) +O(y^41)) );}
%o A322209 {A322200(n,k) = polcoeff( (n+k)*polcoeff( L,n,x),k,y)}
%o A322209 {a(n) = sum(k=0,n, A322200(n-k,k)*2^k )}
%o A322209 for(n=0,40, print1( a(n),", ") )
%Y A322209 Cf. A322200, A322199.
%K A322209 nonn
%O A322209 0,2
%A A322209 _Paul D. Hanna_, Dec 01 2018