Crystal structure of 3 -acetoxy-17 -iodo- 5 -androstenes

The crystal structure of 3β-acetoxy-17α-iodo-Δ⁵-androstene (C₂₁H₃₁O₂I) has been determined from three-dimensional X-ray data and refined to a final reliability index of R = 5.1% for the 2352 observed reflexions. Crystals grown from methylene chloride/methanol are orthorhombic, space group P2₁2₁2₁, with cell constants a = 6.430, b = 10.563, c = 29.369 Å.     

Incomplete cauchy numbers

By using the restricted Stirling numbers and associated Stirling numbers, we introduce two kinds of incomplete Cauchy numbers, which are generalizations that of the classical Cauchy numbers. We also study several arithmetical and combinatorial properties.     

The Riemann surface of the $$r$$ r -Lambert function

Acta Mathematica Hungarica - The $$r$$ -Lambert function is a generalization of the classical Lambert W function which has proven to be useful in physics and other disciplines. In this paper we...     

Synthese und Struktur porphinoider Nickel-Komplexe mit axial orientierten Seitenketten

Synthesis and Structure of Porphine-type Nickel Complexes Containing Axially Oriented Sidechains The structure of (±)-[1, 11-dimethoxy-10 H-HDP (2-)]nickel ( 3 ), the product of the thermodynamically controlled addition of methanol to [6 H-HDP]nickel-bis-(tetrafluoroborate) ( 1 ), was determined by X-ray analysis. The two methoxy groups in 3 are cis-oriented. The syntheses and spectroscopic properties of [1, 11-diethoxy-10 H-HDP (2-)]nickel ( 5 ), [1, 11-dineopentyloxy-10 H-HDP (2-)]nickel ( 6 ) as well as the bridged complexes [1, 11-pentamethylenedioxy-10 H-HDP (2-)]nickel ( 4 ) and [1, 11-(E)-2,3-(dimethyl-2-butenylenedioxy)-10 H-HDP (2-)]nickel ( 7 ) are described. Reaction of 1 with bromide ions or 4-methylpyridine leads to the formation of the corresponding hexacoordinated, paramagnetic complexes dibromo [6 H-HDP]-nickel (II) ( 8 ) and bis (4-methylpyridine)[6 H-HDP]nickel-bis (tetrafluoroborate) ( 9 ).     

Some limit theorems with respect to constrained permutations and partitions

It is a classical result that the ratio of the fixed point free permutations in the whole tends to 1 / e. A generalization of this limit theorem when the cycles must contain at least a given \(m>2\) number of elements is also known. In the present paper we find the limit of the fixed point free permutations with given number of cycles in the whole. In the second part of the paper similar questions are studied when we take partitions in place of permutations.     

Continued fraction expansions for the Lambert $$\varvec{W}$$ function

Aequationes mathematicae - In the first part of the paper we give a new integral representation for the principal branch of the Lambert W function. Then we deduce two continued fraction expansions...     

Hyperharmonic series involving Hurwitz zeta function

We show that the sum of the series formed by the so-called hyperharmonic numbers can be expressed in terms of the Riemann zeta function. These results enable us to reformulate Euler's formula involving the Hurwitz zeta function. In additon, we improve Conway and Guy's formula for hyperharmonic numbers.     

New synthesis of idraparinux,the non-glycosaminoglycan analogue of the antithrombin-binding domain of heparin

Idraparinux, the fully O-sulfated, O-methylated, heparin-related pentasaccharide possessing selective factor Xa inhibitory activity, was prepared by a new synthetic pathway. This route was based on a [2+3] block synthesis utilizing a 6-O-silyl-protected l-idose-containing trisaccharide acceptor, which was glycosylated with a disaccharide donor containing a non-oxidized precursor of the glucuronic acid. The unique strategy of multiple functionalizations at pentasaccharide levels, involving triple methylation followed by oxidation of the glucose and the idose precursors into d-glucuronic and l-iduronic acids in one step, proved to be highly efficient, providing the target pentasaccharide through a 39-step synthesis starting from d-glucose and methyl α-d-glucopyranoside.     

A New Formula for the Bernoulli Polynomials

In this note we show that a seemingly new class of Stirling-type pairs can be applied to produce a new representation of the Bernoulli polynomials at positive rational arguments. A class of generalized harmonic numbers is also investigated, and we point out that these give a new relation for the so-called harmonic polynomials.