Brothers, Harlan J. Pascal’s prism. (English) Zbl 1383.05009 Math. Gaz. 96, No. 536, 213-220 (2012). Summary: Pascal’s triangle is well known for its numerous connections to probability theory, combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series. It also has a deep connection to the base of natural logarithms, \(e\). This link to \(e\) can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object. Cited in 2 Documents MSC: 05A10 Factorials, binomial coefficients, combinatorial functions 28A80 Fractals Keywords:Leibniz harmonic triangle; Leibniz denominator array; multinomial array; fractal representations; tetrahedral embodiment of Pascal’s prism Software:OEIS PDFBibTeX XMLCite \textit{H. J. Brothers}, Math. Gaz. 96, No. 536, 213--220 (2012; Zbl 1383.05009) Full Text: DOI Online Encyclopedia of Integer Sequences: Products across consecutive rows of the denominators of the Leibniz harmonic triangle (A003506). Pascal’s prism (3-dimensional array) read by folded antidiagonal cross-sections: (h+i; h, i-j, j), h >= 0, i >= 0, 0 <= j <= i. Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n. Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n. Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n. References: [1] 1.RossJ. F., Pascal’s legacy, EMBO reports, 5, Special Issue (2004) pp.7-10.10.1038/sj.embor.740005814710175 [2] 2.EdwardsA. W. F., Pascal’s arithmetical triangle: the story of a mathematical idea, Johns Hopkins University Press, Baltimore (2002) pp. xiii, 1, 27, 34-37. · Zbl 1032.01013 [3] 3.GullbergJ., Mathematics: from the birth of numbers, W. W. Norton and Company, New York (1997) p. 141. · Zbl 0873.00001 [4] 4.WeissteinE. W., Pascal’s Triangle from MathWorld – a Wolfram Web Resource. http://mathworld.wolfram.com/PascalsTriangle.html [5] 5.BrothersH. J., Pascal’s triangle: The hidden stor-e, Math. Gaz. (March 2012), pp. 145-148. [6] 6.SloaneN. J. A., Sequence A001142. http://oeis.org/A001142 [7] 7.BrothersH. J. and KnoxJ. A., New closed-form approximations to the logarithmic constant e, The Mathematical Intelligencer20 (1998) pp. 25-29.10.1007/BF03025225 · Zbl 0956.11028 [8] 8.KnoxJ. A. and BrothersH. J., Novel series-based approximations to e, College Mathematics Journal30 (1999) pp. 269-275.10.2307/2687664 · Zbl 0995.11519 [9] 9.BrothersH. J., Sequence A168510. http://oeis.org/A168510 [10] 10.WeissteinE. W., Leibniz Harmonic Triangle from MathWorld – a Wolfram Web Resource. http://mathworld.wolfram.com/LeibnizHarmonicTriangle.html [11] 11.RichardsonL. F. and GauntJ. A., The deferred approach to the limit, Philosophical Transactions of the Royal Society of London 226a (1927) pp. 299-361. · JFM 53.0432.02 [12] 12.BrothersH. J., Sequence A191510. https://oeis.org/A191510 [13] 13.AdamsonG. W., Sequence A132818. http://oeis.org/A132818 [15] 15.PickoverC. A., Computers, pattern, chaos, and beauty: graphics from an unseen world, St. Martin’s Press, New York (1990) pp. 173-185. [16] 16.GuyR. K., The second strong law of small numbers, Mathematics Magazine63 (1990), pp. 3-20.10.2307/2691503 · Zbl 0708.11001 [17] 17.BardzellM. and ShannonK., The PascGalois project: visualizing abstract algebra, Focus22 (2002) pp. 4-5. [18] 18.FrameM. L. and MandelbrotB. B., Fractals, graphics, and mathematics education, Cambridge University Press (2002). · Zbl 1007.00010 [19] 19.FrameM. L. and NegerN., Fractal tetrahedra: What’s left in, what’s left out, and how to build one in four dimensions, Computers & Graphics32 (2008) pp. 371-381.10.1016/j.cag.2007.12.001 [20] 20.WeissteinE. W., Tetrix from Math World – a Wolfram Web Resource. http://mathworld.wolfram.com/Tetrix.html [21] 21.FrameM. L. and NegerN., Dimensions and the probability of finding odd numbers in Pascal’s triangle and its relatives, Computers & Graphics34 (2010) pp. 158-166.10.1016/j.cag.2009.10.002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.