Binomial coefficients, Catalan numbers and Lucas quotients

Let $p$ be an odd prime and let $a,m$ be integers with $a>0$ and $m \not\equiv0\pmod p$. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ mod $p^2$ for $d=0,1$; for example, $$\sum_{k=0}^{p^a-1}\frac{\binom{2k}k}{m^k}\equiv\left(\frac{m^2-4m}{p^a}\right)+\left(\frac{m^2-4m}{p^{a-1}}\right)u_{p-(\frac{m^2-4m}{p})}\pmod{p^2},$$ where $(-)$ is the Jacobi symbol, and $\{u_n\}_{n\geqslant0}$ is the Lucas sequence given by $u_0=0$, $u_1=1$ and $u_{n+1}=(m-2)u_n-u_{n-1}$ for $n=1,2,3,\ldots$. As an application, we determine $\sum_{0<k<p^a,\, k\equiv r\pmod{p-1}}C_k$ modulo $p^2$ for any integer $r$, where $C_k$ denotes the Catalan number $\binom{2k}k/(k+1)$. We also pose some related conjectures.Comment: 24 pages. Correct few typo

Effects of the calcium channel antagonist mibefradil on haemodynamic and morphological parameters in myocardial infarction-induced cardiac failure in rats

Calcium channel antagonists (CCA) have been proposed for the prevention of cardiac events after myocardial infarction (MI). Mibefradil is a CCA featuring a selective blockade of T-type Ca2(+)-channels. The aim of the study was to characterize the effects of mibefradil on haemodynamic and morphological parameters in a model of postMI chronic heart failure and to establish the "therapeutic window" for the start of therapy. MI was induced by permanent ligation of the left coronary artery in male normotensive Wistar rats. Animals were assigned to placebo- or mibefradil-treated (10 mg/kg/day p.o.) groups as follows: (1) sham operation; (2) MI placebo treatment; (3) 7 days preMI start of treatment; (4) 3 h postMI start of treatment; (5) 24 h postMI start of treatment; (6) 3 days postMI start of treatment; (7) 7 days postMI start of treatment. Treatment was continued for 6 weeks postMI. At this time point, mean arterial blood pressure (MAP), heart rate, left ventricular enddiastolic pressure (LVEDP) and contraction force (dP/dtmax) were measured in conscious rats at baseline and after methoxamine (MEX; 0.5-1.0 mg/h i.v.) stimulation to increase afterload. The hearts were subjected to histological determination of infarct size (IS), infarct length (IL), noninfarcted length (NL), left ventricular circumference (LVC), inner LV-diameter (LVD) and septal thickness (ST). Six weeks after MI, MAP was lowered, LVEDP increased and dP/dtmax reduced. Mibefradil treatment increased basal MAP in groups 3-5 compared to the placebo-treated MI group. Under mibefradil, LVEDP was reduced at baseline in groups 3-6 and, after MEX, in all groups. dP/dtmax was increased in groups 3-4 at baseline and after MEX. In the placebo-treated MI group, the infarcted area was 39% of the LV and heart weight, LVD and LVC were increased. Heart weights of mibefradil-treated rats (groups 3-6) did not differ from those of the placebo-treated group. Early onset of treatment with mibefradil reduced IS and IL and increased NL in groups 3-4. LVD and LVC were decreased in group 3 only. ST was increased in groups 3-5. Chronic treatment with mibefradil exerts beneficial actions on cardiac structure and performance in postMI cardiac failure in rats, especially when the onset of treatment is either prior to or within hours after the acute ischemic even