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%I M1667 #36 Sep 22 2019 08:02:56
%S 1,1,2,6,23,107,586,3690,26245,207997,1817090,17345358,179595995,
%T 2004596903,23992185226,306497734962,4162467826729,59882101858777,
%U 909688617178178,14551535460258966,244477068964113407
%N Number of standard paths of length n in composition poset.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H F. Bergeron, M. Bousquet-Mélou and S. Dulucq, <a href="/A007555/a007555.pdf">Standard paths in the composition poset</a>, Preprint. (Annotated scanned copy)
%H F. Bergeron, M. Bousquet-Mélou and S. Dulucq, <a href="http://www.labmath.uqam.ca/~annales/volumes/19-2/PDF/139-151.pdf">Standard paths in the composition poset</a>, Ann. Sci. Math. Quebec, 19 (1995), no. 2, 139-151.
%H <a href="/index/Pos#posets">Index entries for sequences related to posets</a>
%F E.g.f.: exp(-x)/(cosh(x/sqrt(2)) - sqrt(2)*sinh(x/sqrt(2)))^2.
%F a(n) is asymptotic to n!(c1(n+1+c2)+c2)/((sqrt(2)c2)^n*c1^c1*c2^2) where c1=1+sqrt(2), c2=log(c1).
%F G.f.: 1/Q(0), where Q(k) = 1 - x*(2*k+1) - x^2*(k+1)*(k+2)/2/Q(k+1) ; (continued fraction). - _Sergei N. Gladkovskii_, Sep 29 2013
%t terms = 21;
%t 1/(1 - x + ContinuedFractionK[-(k(k+1)/2)x^2, 1-(2k+1) x, {k, 1, terms/2 // Floor}]) + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Sep 22 2019, after _Sergei N. Gladkovskii_ *)
%o (PARI) a(n)=local(X,w); if(n<0,0,X=x+x*O(x^n); w=quadgen(8); n!*polcoeff(exp(-X)/(cosh(X/w)-w*sinh(X/w))^2,n))
%o (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(intformal(1/(1-intformal(1/cosh((x+x*O(x^n))/quadgen(8))^2)))),n))
%K nonn
%O 0,3
%A _Simon Plouffe_
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