A007555 Number of standard paths of length n in composition poset. (Formerly M1667)

1, 1, 2, 6, 23, 107, 586, 3690, 26245, 207997, 1817090, 17345358, 179595995, 2004596903, 23992185226, 306497734962, 4162467826729, 59882101858777, 909688617178178, 14551535460258966, 244477068964113407

Offset

0,3

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=0..20.

F. Bergeron, M. Bousquet-Mélou and S. Dulucq, Standard paths in the composition poset, Preprint. (Annotated scanned copy)

F. Bergeron, M. Bousquet-Mélou and S. Dulucq, Standard paths in the composition poset, Ann. Sci. Math. Quebec, 19 (1995), no. 2, 139-151.

Index entries for sequences related to posets

Formula

E.g.f.: exp(-x)/(cosh(x/sqrt(2)) - sqrt(2)*sinh(x/sqrt(2)))^2.

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).

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

Mathematica

terms = 21;
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 *)

Prog

(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))
(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))

Crossrefs

Sequence in context: A200403 A113226 A071075 * A101053 A155857 A071076

Adjacent sequences: A007552 A007553 A007554 * A007556 A007557 A007558

Keyword

nonn

Author

Simon Plouffe

Status

approved