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A007555 Number of standard paths of length n in composition poset.
(Formerly M1667)
1
1, 1, 2, 6, 23, 107, 586, 3690, 26245, 207997, 1817090, 17345358, 179595995, 2004596903, 23992185226, 306497734962, 4162467826729, 59882101858777, 909688617178178, 14551535460258966, 244477068964113407 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 9 17:46 EST 2022. Contains 358703 sequences. (Running on oeis4.)