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%I #15 Apr 18 2017 07:04:12
%S 1,1,3,9,32,117,460,1850,7680,32467,139631,608280,2679881,11916492,
%T 53417407,241118245,1095030281,4999847240,22938576846,105690791104,
%U 488860041249,2269080989966,10565660913109,49340702712984,231031730790560
%N A simple grammar.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=843">Encyclopedia of Combinatorial Structures 843</a>
%F G.f. A(x) satisfies [from _Paul D. Hanna_, Oct 16 2013]:
%F (1) A(x) = exp( Sum_{n>=1} x^n/n * A(x^n)/(1 - x^n*A(x^n)) );
%F (2) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{d|n} d*A(x^(n/d))^d ).
%e G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 32*x^4 + 117*x^5 + 460*x^6 +...
%e where
%e log(A(x)) = x*A(x)/(1-x*A(x)) + x^2/2*A(x^2)/(1-x^2*A(x^2)) + x^3/3*A(x^3)/(1-x^3*A(x^3)) + x^4/4*A(x^4)/(1-x^4*A(x^4)) + x^5/5*A(x^5)/(1-x^5*A(x^5)) + x^6/6*A(x^6)/(1-x^6*A(x^6)) +... - _Paul D. Hanna_, Oct 16 2013
%p spec := [S,{C=Sequence(B,1 <= card),S=Set(C),B=Prod(Z,S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(k=1, n, x^k/k*subst(A,x,x^k)/(1-x^k*subst(A,x,x^k+x*O(x^n)) )))); polcoeff(A,n)} \\ _Paul D. Hanna_, Oct 16 2013
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(k=1, n, x^k/k*sumdiv(k, d, d*subst(A,x,x^(k/d) +x*O(x^n))^d)))); polcoeff(A,n)} \\ _Paul D. Hanna_, Oct 16 2013
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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