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

1, 1, 1, 2, 4, 9, 17, 36, 71, 143, 284, 573, 1140, 2287, 4568, 9138, 18272, 36559, 73098, 146216, 292413, 584836, 1169657, 2339353, 4678655, 9357356, 18714673, 37429377, 74858706, 149717506, 299434883, 598869895, 1197739689, 2395479446, 4790958784, 9581917760, 19163835261, 38327670814, 76655341388, 153310682944, 306621365618, 613242731721, 1226485462828, 2452970926285, 4905941852039, 9811883704440, 19623767408346, 39247534817726, 78495069634129

Offset: 0

Author

Paul D. Hanna, Mar 26 2016

Keywords

• nonn

Comments

Compare g.f. to the identity: x = Sum_{n>=1} x^n/(1+x)^n.

Examples

			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 +...
where
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) *...
RELATED SERIES.
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 +...
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 +...
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 +...
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 +...
A(x^5/(1+x)^5) = 1 + x^5 - 5*x^6 + 15*x^7 - 35*x^8 + 70*x^9 - 125*x^10 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A117209.

Programs

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]}
for(n=0,40,print1(a(n),", "))

Formula

G.f. satisfies: (1-x)/(1-2*x) = Product_{n>=1} A(x^n).

a(n) ~ c * 2^n, where c = 0.2788705076091492504414859194394933690344541628... . - Vaclav Kotesovec, Apr 02 2016