A079144 -id:A079144 - cp's OEIS frontend

A208732 Sequence related to Kashaev's invariant for the (9,2)-torus knot.

1, 4, 36, 664, 21276, 1050664, 73939356, 7024817944, 866058563196, 134419597387144, 25644210185987676, 5897980691609567224, 1609292585008090909116, 513950106691675965931624, 189914985024774644611299996

Offset: 0

Peter Bala, Mar 01 2012

This is sequence b_n(9) in Table 2 of Hikami 2003.

K. Hikami, Volume Conjecture and Asymptotic Expansion of q-Series, Experimental Mathematics Vol. 12, Issue 3 (2003).

Cf. A079144, A208681, A208730, A208731, A208735.

a(n) = (49/72)^n*sum {k = 0..n} binomial(n,k)*A208681(k+1)/49^k.

Conjectural S-fraction for the o.g.f.: 1/(1-4*x/(1-5*x/(1-17*x/(1-19*x/(1-...-1/2*n*(9*n-1)*x/(1-1/2*n*(9*n+1)*x/(1- ....

A301871 Number of N- and bowtie-free posets with n elements.

Original entry on oeis.org

1, 2, 5, 14, 40, 121, 373, 1184, 3823, 12554, 41733, 140301, 475934, 1627440, 5602983, 19406703, 67574371, 236409625, 830582851, 2929246932, 10366380583, 36801225872, 131021870786, 467701875135, 1673584553886, 6002046468815, 21570135722058, 77668429499325, 280167079428684, 1012323004985313

Offset: 1

Stephan Wagner, Mar 28 2018

The number of n-element posets that do not include the two 4-element posets "N" and "bowtie" as induced subposets.

Stephan Wagner, Table of n, a(n) for n = 1..100
T. Hasebe and S. Tsujie, Order quasisymmetric functions distinguish rooted trees, arXiv:1610.03908 [math.CO], 2016-2017.
T. Hasebe and S. Tsujie, Order quasisymmetric functions distinguish rooted trees, Journal of Algebraic Combinatorics 46 (2017), 499-515.
V. Razanajatovo Misanantenaina and S. Wagner, A Tutte-like polynomial for rooted trees and specific posets, arXiv:1803.09623 [math.CO], 2018.

Cf. A000112, A003430, A079144, A079146 for related sequences regarding the enumeration of unlabeled posets.

V=1;Do[V = Normal[Series[(1 - x) Exp[Sum[(2 x^m - x^(2 m)) (V /. x -> x^m)/m, {m, 1, n}]], {x, 0, n}]], {n, 1, 20}]; Table[Coefficient[V,x,n],{n, 1, 20}]

G.f. V(x) = 1 + x + 2x + 5x^2 + ... satisfies V(x) = (1-x)exp[sum_{m >=1} (2x^m-x^(2m))V(x^m)/m] (see Razanajatovo Misanantenaina/Wagner).

A187803 E.g.f.: Sum_{n>=0} Product_{k=1..n} (1 - exp(-nkx)).

Original entry on oeis.org

1, 1, 15, 829, 113487, 31137061, 15015039495, 11636995485949, 13584094722071007, 22735343245138118101, 52487807127760090483575, 162018777092079952134169069, 651747862300297714019151918927, 3344015911143306355676226376118341, 21488215819992049616143504500848490855

Offset: 0

Paul D. Hanna, Jan 06 2013

nonn

Compare to the e.g.f. of A079144, enumerating certain labeled interval orders:
Sum_{n>=0} Product_{k=1..n} (1 - exp(-k*x)).
Also compare to the e.g.f. of A220181: Sum_{n>=0} (1 - exp(-n*x))^n.

			E.g.f.: A(x) = 1 + x + 15*x^2/2! + 829*x^3/3! + 113487*x^4/4! +...
where
A(x) = 1 + (1-exp(-x)) + (1-exp(-2*1*x))*(1-exp(-2*2*x)) + (1-exp(-3*1*x))*(1-exp(-3*2*x))*(1-exp(-3*3*x)) + (1-exp(-4*1*x))*(1-exp(-4*2*x))*(1-exp(-4*3*x))*(1-exp(-4*4*x)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..140

Cf. A079144, A220181.

{a(n)=n!*polcoeff(sum(m=0, n, prod(k=1,m,(1-exp(-m*k*x+x*O(x^n)))) ), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * d^n * (n!)^3 / sqrt(n), where d = 2.426663845780394275167988381..., c = 0.504146101604802096078745... . - Vaclav Kotesovec, Nov 03 2014

cp's OEIS Frontend

A208732 Sequence related to Kashaev's invariant for the (9,2)-torus knot.

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A301871 Number of N- and bowtie-free posets with n elements.

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Formula

A187803 E.g.f.: Sum_{n>=0} Product_{k=1..n} (1 - exp(-nkx)).

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PARI

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cp's OEIS Frontend

A208732 Sequence related to Kashaev's invariant for the (9,2)-torus knot.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A301871 Number of N- and bowtie-free posets with n elements.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A187803 E.g.f.: Sum_{n>=0} Product_{k=1..n} (1 - exp(-n*k*x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A187803 E.g.f.: Sum_{n>=0} Product_{k=1..n} (1 - exp(-nkx)).