Woon's tree and sums over compositions

This article studies sums over all compositions of an integer. We derive a generating function for this quantity, and apply it to several special functions, including various generalized Bernoulli numbers. We connect composition sums with a recursive tree introduced by S.G. Woon and extended by P. Fuchs under the name "general PI tree", in which an output sequence $\{x_n\}$ is associated to the input sequence $\{g_n\}$ by summing over each row of the tree built from $\{g_n\}$. Our link with the notion of compositions allows to introduce a modification of Fuchs' tree that takes into account nonlinear transforms of the generating function of the input sequence. We also introduce the notion of \textit{generalized sums over compositions}, where we look at composition sums over each part of a composition

A Continuous Analogue of Lattice Path Enumeration: Part II

Here are exhibited some additional results about the continuous binomial coefficients as introduced by L. Cano and R. Diaz in [1].Comment: second versio

Settling some sum suppositions

International audienc

Six end-stage renal disease patients benefited from first non-simultaneous single center 6-way kidney exchange transplantation in India

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Expanding the donor pool for kidney transplantation in India

The best and most cost-effective treatment for end-stage renal disease patients is living donor (LD) renal transplantation. It has survival benefit compared to deceased donor (DD) kidney transplant (DDKT) and long-term dialysis and provides a better quality of life. Efficient and effective kidney allocation methods are increasingly necessary to address the problem of organ scarcity. The use of kidney paired donation transplant has increased access to LD kidney transplantation (LDKT) with outstanding results. ABO-incompatible kidney transplantation (KT) and desensitization protocol can expand the donor pool, but as integral to any aggressive immunosuppression protocol, they are associated with increased risk of infection and malignancy. Given the widespread organ shortage, DDKT from donors with sepsis, donors who died from snakebite or acute kidney injury, controlled donation after cardiac death, older donors, can be considered for KT with an acceptable outcome. The acceptable outcome can be achieved with dual KT using kidneys from expanded criteria donors in older population. Dual KT from pediatric donors to adult recipients or from adult marginal DDs is a promising way to expand the donor pool. Carefully selected donor with HIV, HCV, and HBV positivity is not a contraindication for living kidney donation. Careful and meticulous selection of patient and donor is essential for successful outcome. Affordable or free transplantation is other way to increase transplantation rate in developing country. The community support can make transplantation available free to the poor patients under community-government partnership. Various steps should be taken to promote LDKT and DDKT program