|
%I #43 Nov 27 2023 03:17:47
%S 1,2,2,6,4,6,20,10,12,20,70,28,28,40,70,252,84,72,90,140,252,924,264,
%T 198,220,308,504,924,3432,858,572,572,728,1092,1848,3432,12870,2860,
%U 1716,1560,1820,2520,3960,6864,12870,48620,9724,5304,4420,4760,6120,8976
%N Triangle T(n,k) = (2*k)!*(2*n)!/(k!*n!*(k+n)!) with k=0..n, read by rows.
%C This is a companion to the triangle A068555.
%C Row sum is 2*A132310(n-1) + A000984(n) for n>0, where A000984(n) = T(n,0) = T(n,n). Also:
%C T(n,1) = -A002420(n+1).
%C T(n,2) = A002421(n+2).
%C T(n,3) = -A002422(n+3) = 2*A007272(n).
%C T(n,4) = A002423(n+4).
%C T(n,5) = -A002424(n+5).
%C T(n,6) = A020923(n+6).
%C T(n,7) = -A020925(n+7).
%C T(n,8) = A020927(n+8).
%C T(n,9) = -A020929(n+9).
%C T(n,10) = A020931(n+10).
%C T(n,11) = -A020933(n+11).
%D Umberto Scarpis, Sui numeri primi e sui problemi dell'analisi indeterminata in Questioni riguardanti le matematiche elementari, Nicola Zanichelli Editore (1924-1927, third edition), page 11.
%D J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 103.
%H Alexander Borisov, <a href="https://arxiv.org/abs/math/0505167">Quotient singularities, integer ratios of factorials and the Riemann Hypothesis</a>, arXiv:math/0505167 [math.NT], 2005; International Mathematics Research Notices, Vol. 2008, Article ID rnn052, page 2 (Theorem 2).
%H Ira Gessel, <a href="http://people.brandeis.edu/~gessel/homepage/slides/int-quot.pdf">Integer quotients of factorials and algebraic multivariable hypergeometric series</a>, MIT Combinatorics Seminar, September 2011 (slides).
%H Hans-Christian Herbig and Mateus de Jesus Gonçalves, <a href="https://arxiv.org/abs/2311.13604">On the numerology of trigonometric polynomials</a>, arXiv:2311.13604 [math.HO], 2023.
%H Kevin Limanta and Norman Wildberger, <a href="https://arxiv.org/abs/2108.10191">Super Catalan Numbers, Chromogeometry, and Fourier Summation over Finite Fields</a>, arXiv:2108.10191 [math.CO], 2021. See Table 1 p. 2 where terms are shown as an array.
%e Triangle begins:
%e 1;
%e 2, 2;
%e 6, 4, 6;
%e 20, 10, 12, 20;
%e 70, 28, 28, 40, 70;
%e 252, 84, 72, 90, 140, 252;
%e 924, 264, 198, 220, 308, 504, 924;
%e 3432, 858, 572, 572, 728, 1092, 1848, 3432;
%e 12870, 2860, 1716, 1560, 1820, 2520, 3960, 6864, 12870;
%e 48620, 9724, 5304, 4420, 4760, 6120, 8976, 14586, 25740, 48620;
%e ...
%e Sum_{k=0..8} T(8,k) = 12870 + 2860 + 1716 + 1560 + 1820 + 2520 + 3960 + 6864 + 12870 = 2*A132310(7) + A000984(8) = 2*17085 + 12870 = 47040.
%t Flatten[Table[Table[(2 k)! ((2 n)!/(k! n! (k + n)!)), {k, 0, n}], {n, 0, 9}]]
%o (Magma)
%o [Factorial(2*k)*Factorial(2*n)/(Factorial(k)*Factorial(n)*Factorial(k+n)): k in [0..n], n in [0..9]];
%Y Cf. A000984, A002420-A020933, A068555, A132310.
%K nonn,tabl,look,easy
%O 0,2
%A _Bruno Berselli_, Apr 27 2012
|
