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A268649 G.f. A(x) satisfies: 1/(1-x) = Product_{n>=1} A( x^n/(1+x)^n ).

%I A268649 #8 Apr 02 2016 12:42:02
%S A268649 1,1,1,2,4,9,17,36,71,143,284,573,1140,2287,4568,9138,18272,36559,
%T A268649 73098,146216,292413,584836,1169657,2339353,4678655,9357356,18714673,
%U A268649 37429377,74858706,149717506,299434883,598869895,1197739689,2395479446,4790958784,9581917760,19163835261,38327670814,76655341388,153310682944,306621365618,613242731721,1226485462828,2452970926285,4905941852039,9811883704440,19623767408346,39247534817726,78495069634129
%N A268649 G.f. A(x) satisfies: 1/(1-x) = Product_{n>=1} A( x^n/(1+x)^n ).
%C A268649 Compare g.f. to the identity: x = Sum_{n>=1} x^n/(1+x)^n.
%H A268649 Vaclav Kotesovec, <a href="/A268649/b268649.txt">Table of n, a(n) for n = 0..200</a>
%F A268649 G.f. satisfies: (1-x)/(1-2*x) = Product_{n>=1} A(x^n).
%F A268649 a(n) ~ c * 2^n, where c = 0.2788705076091492504414859194394933690344541628... . - _Vaclav Kotesovec_, Apr 02 2016
%e A268649 G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 17*x^6 + 36*x^7 + 71*x^8 + 143*x^9 + 284*x^10 + 573*x^11 + 1140*x^12 +...
%e A268649 where
%e A268649 1/(1-x) = A(x/(1+x)) * A(x^2/(1+x)^2) * A(x^3/(1+x)^3) * A(x^4/(1+x)^4) * A(x^5/(1+x)^5) *...
%e A268649 RELATED SERIES.
%e A268649 A(x/(1+x)) = 1 + x + x^3 + 2*x^5 - 4*x^6 + 14*x^7 - 35*x^8 + 86*x^9 - 191*x^10 +...
%e A268649 A(x^2/(1+x)^2) = 1 + x^2 - 2*x^3 + 4*x^4 - 8*x^5 + 17*x^6 - 38*x^7 + 88*x^8 +...
%e A268649 A(x^3/(1+x)^3) = 1 + x^3 - 3*x^4 + 6*x^5 - 9*x^6 + 9*x^7 - 26*x^9 + 72*x^10 +...
%e A268649 A(x^4/(1+x)^4) = 1 + x^4 - 4*x^5 + 10*x^6 - 20*x^7 + 36*x^8 - 64*x^9 + 120*x^10 +...
%e A268649 A(x^5/(1+x)^5) = 1 + x^5 - 5*x^6 + 15*x^7 - 35*x^8 + 70*x^9 - 125*x^10 +...
%o A268649 (PARI) {a(n) = my(A=[1,1],X=x+x*O(x^n)); for(i=1,n, A=concat(A,0); A[#A] = 1 - Vec( prod(k=1,#A, subst(Ser(A),x,x^k/(1+X)^k)) )[#A] );A[n+1]}
%o A268649 for(n=0,40,print1(a(n),", "))
%Y A268649 Cf. A117209.
%K A268649 nonn
%O A268649 0,4
%A A268649 _Paul D. Hanna_, Mar 26 2016