A134242 Numerators of certain constants c_n = A180609(n)/n! related to Hurwitz numbers.

1, -1, 1, -2, 11, -3, -11, 29, 493, -2711, -12406, 2636317, -10597579, -439018457, 1165403153, 118734633647, -105428488301, -4070802683898, 1695077946695371, 56532812889378221, -252968859037883917, -425882179787933647571, 123624959518930226565553, 32729394708071881944913, -5814212300444136523052695

Offset: 1

Author

N. J. A. Sloane, Jan 30 2008

Keywords

• sign
• frac
• easy

Comments

Manetti-Ricciardi refer to the c_n as Koszul numbers.

Examples

			The fractions are 1, -1/2, 1/2, -2/3, 11/12, -3/4, -11/6, 29/4, 493/12, -2711/6, -12406/15,  2636317/60, -10597579/120, -439018457/60, 1165403153/20, 118734633647/60, ...
		

Links

• M Manetti, G Ricciardi, Universal Lie formulas for higher antibrackets, arXiv preprint arXiv:1509.09032 [math.QA], 2015-2016.
• S. Shadrin and D. Zvonkine, Changes of variables in ELSV-type formulas, Michigan Mathematical Journal, vol. 55 (2007), 209-228.
• D. Zvonkine, Home Page

Crossrefs

Cf. A134243, A180609.

Programs

Mathematica

K[1] = 1;
K[n_] := K[n] = -2/((n+2)(n-1)) Sum[StirlingS2[n+1, i] K[i], {i, 1, n-1}];
Table[Numerator[K[n]], {n, 1, 25}] (* Jean-François Alcover, Jul 26 2018 *)

Formula

Manetti-Ricciardi Theorem 4.4 give a recurrence for the c_n in terms of Stirling numbers.

Extensions

More terms from Manetti-Ricciardi added by N. J. A. Sloane, May 25 2016