This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A068555 #70 Sep 19 2023 06:35:19
%S A068555 1,2,2,6,2,6,20,4,4,20,70,10,6,10,70,252,28,12,12,28,252,924,84,28,20,
%T A068555 28,84,924,3432,264,72,40,40,72,264,3432,12870,858,198,90,70,90,198,
%U A068555 858,12870,48620,2860,572,220,140,140,220,572,2860,48620,184756,9724
%N A068555 Triangle read by rows in which row n contains (2i)!*(2j)!/(i!*j!*(i+j)!) for i + j = n, i = 0..n.
%C A068555 One of three infinite families of integral factorial ratio sequences of height 1 (see Bober, Theorem 1.2). The other two are A007318 and A046521. A related table is A182073. - _Peter Bala_, Apr 10 2012
%D A068555 R. K. Guy and Cal Long, Email to N. J. A. Sloane, Feb 22, 2002.
%D A068555 Peter J. Larcombe and David R. French, On the integrality of the Catalan-Larcombe-French sequence 1,8,80,896,10816,.... Proceedings of the Thirty-second Southeastern International Conference on Combinatorics, Graph Theory and Computing (Baton Rouge, LA, 2001). Congr. Numer. 148 (2001), 65-91. MR1887375
%D A068555 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.
%H A068555 Vincenzo Librandi, <a href="/A068555/b068555.txt">Rows n = 0..100, flattened</a>
%H A068555 J. W. Bober, <a href="http://arxiv.org/abs/0709.1977">Factorial ratios, hypergeometric series, and a family of step functions</a>, 2007, arXiv:0709.1977v1 [math.NT]; J. London Math. Soc. (2) 79 (2009), 422-444.
%H A068555 B. Buca and T. Prosen, <a href="http://arxiv.org/abs/1509.04911">Connected correlations, fluctuations and current of magnetization in the steady state of boundary driven XXZ spin chains</a>, arXiv preprint arXiv:1509.04911 [cond-mat.stat-mech], 2015.
%H A068555 Ira Gessel, <a href="https://people.brandeis.edu/~gessel/homepage/slides/nonneg.pdf">Rational functions with nonnegative power series</a>, (slides).
%H A068555 Ira Gessel, <a href="https://people.brandeis.edu/~gessel/homepage/papers/superballot.pdf">Super ballot numbers</a>.
%H A068555 Thomas M. Richardson, <a href="http://arxiv.org/abs/1405.6315">The Reciprocal Pascal Matrix</a>, arXiv preprint arXiv:1405.6315 [math.CO], 2014.
%H A068555 Thomas M. Richardson, <a href="http://arxiv.org/abs/1410.5880">The Super Patalan Numbers</a>, arXiv preprint arXiv:1410.5880 [math.CO], 2014.
%H A068555 Thomas M. Richardson, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Richardson/rich2.html">The Super Patalan Numbers</a>, J. Int. Seq. 18 (2015) # 15.3.3.
%H A068555 Thomas M. Richardson, <a href="http://arxiv.org/abs/1609.01193">The three 'R's and Dual Riordan Arrays</a>, arXiv:1609.01193 [math.CO], 2016.
%H A068555 R. Sprugnoli, <a href="https://web.archive.org/web/20120127210623/http://www.dsi.unifi.it/~resp/GouldBK.pdf">Riordan array proofs of identities in Gould's book</a>.
%F A068555 The square array defined by f := (a, b)->add(binomial(2*a, k)*binomial(2*b, a+b-k)*(-1)^(a+b-k), k=0..2*a); and read by antidiagonals gives a signed version. See Sprugnoli, 3.38.
%F A068555 Let f(x) = 1/sqrt(1 - 4*x) denote the o.g.f for A000984. The o.g.f. for this table is (f(x) + f(y))*f(x)*f(y)*(1/(1 + f(x)*f(y))) = (1 + 2*x + 6*x^2 + 20*x^3 + ...) + (2 + 2*x + 4*x^2 + 10*x^3 + ...)*y + (6 + 4*x + 6*x^2 + 12*x^3 + ...)*y^2 + .... - _Peter Bala_, Apr 10 2012
%F A068555 T(n,0) = A000984(n), T(n,k) = 4*T(n-1,k-1) - T(n,k-1) for k = 1..n. - _Philippe Deléham_, Mar 10 2014
%e A068555 From _Bruno Berselli_, Apr 27 2012: (Start)
%e A068555 Triangle begins:
%e A068555 1;
%e A068555 2, 2;
%e A068555 6, 2, 6;
%e A068555 20, 4, 4, 20;
%e A068555 70, 10, 6, 10, 70;
%e A068555 252, 28, 12, 12, 28, 252;
%e A068555 924, 84, 28, 20, 28, 84, 924;
%e A068555 3432, 264, 72, 40, 40, 72, 264, 3432;
%e A068555 12870, 858, 198, 90, 70, 90, 198, 858, 12870;
%e A068555 48620, 2860, 572, 220, 140, 140, 220, 572, 2860, 48620;
%e A068555 184756, 9724, 1716, 572, 308, 252, 308, 572, 1716, 9724, 184756; ...
%e A068555 (End)
%e A068555 T(4,0) = A000984(4) = 70, T(4,1) = 4*20 - 70 = 10, T(4,2) = 4*4 - 10 = 6, T(4,3) = 4*4 - 6 = 10, T(4,4) = 4*20 - 10 = 70. - _Philippe Deléham_, Mar 10 2014
%p A068555 A068555 := proc(n,i)
%p A068555 j := n-i ;
%p A068555 (2*i)!*(2*j)!/(i!*j!*(i+j)!) ;
%p A068555 end proc: # _R. J. Mathar_, May 31 2016
%t A068555 Flatten[ Table[ Table[ (2i)!*(2(n - i))!/(i!*(n - i)!*n!), {i, 0, n}], {n, 0, 9}]]
%o A068555 (PARI) a(n,k)=if(n<0 || k<0,0,(2*n)!*(2*k)!/n!/k!/(n+k)!);
%o A068555 (Magma) [Factorial(2*i)*Factorial(2*(n-i))/(Factorial(i)*Factorial(n)*Factorial(n-i)): i in [0..n], n in [0..10]]; // _Bruno Berselli_, Apr 27 2012
%Y A068555 Apart perhaps from signs, diagonals give A000984, A002420, A078718.
%Y A068555 Cf. A007318, A046521, A182073.
%Y A068555 Cf. A182411, A082590 (row sums).
%K A068555 nonn,tabl,easy,nice
%O A068555 0,2
%A A068555 _N. J. A. Sloane_, Mar 23 2002