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G.f.: A(x) = exp( Sum_{n>=1} A056045(n)/n*x^n ), where A056045(n) = Sum_{d|n} binomial(n,d).
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%I #14 Jan 15 2019 08:18:27

%S 1,1,2,3,6,8,18,23,49,73,145,194,474,611,1331,2027,4393,5919,14736,

%T 19415,46487,68504,156618,212055,560380,739165,1833012,2657837,

%U 6513367,8743208,23649777,31140300,81276046,114962333,293600318,391926154

%N G.f.: A(x) = exp( Sum_{n>=1} A056045(n)/n*x^n ), where A056045(n) = Sum_{d|n} binomial(n,d).

%H Seiichi Manyama, <a href="/A110448/b110448.txt">Table of n, a(n) for n = 0..3337</a>

%F G.f.: A(x) = Product_{n>=1} (1/x)*Series_Reversion( x/(1 + x^n) ); equivalently, G.f.: A(x) = Product_{n>=1} G(x^n,n) where G(x,n) = 1 + x*G(x,n)^n.

%F a(n) ~ c * 2^n / n^(3/2), where c = 2.8176325363130737043447... if n is even and c = 1.784372019603712867208... if n is odd. - _Vaclav Kotesovec_, Jan 15 2019

%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 18*x^6 +...

%e where A(x) = exp( Sum_{n>=1} A056045(n)/n*x^n ), or

%e A(x) = exp(x + 3/2*x^2 + 4/3*x^3 + 11/4*x^4 + 6/5*x^5 +...).

%e The g.f. can also be expressed as the product:

%e A(x) = 1/(1-x)*G000108(x^2)*G001764(x^3)*G002293(x^4)*G002294(x^5)*...

%e where the functions are g.f.s of well-known sequences:

%e G000108(x) = 1 + x*G000108(x)^2 = g.f. of A000108 ;

%e G001764(x) = 1 + x*G001764(x)^3 = g.f. of A001764 ;

%e G002293(x) = 1 + x*G002293(x)^4 = g.f. of A002293 ;

%e G002294(x) = 1 + x*G002294(x)^5 = g.f. of A002294 ; etc.

%o (PARI) {a(n)=polcoeff(exp(x*Ser(vector(n,m, sumdiv(m,d,binomial(m,d))/m))+x*O(x^n)),n)}

%o (PARI) {a(n)=polcoeff(prod(m=1,n,1/x*serreverse(x/(1+x^m +x*O(x^n)))),n)}

%Y Cf. A056045, A174461, A206290.

%Y Cf. A000108, A001764, A002293, A002294, A002295.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 20 2005, Nov 10 2007