A068555 - OEIS

%I #70 Sep 19 2023 06:35:19

%S 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 28,84,924,3432,264,72,40,40,72,264,3432,12870,858,198,90,70,90,198,

%U 858,12870,48620,2860,572,220,140,140,220,572,2860,48620,184756,9724

%N Triangle read by rows in which row n contains (2i)!*(2j)!/(i!*j!*(i+j)!) for i + j = n, i = 0..n.

%C 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 R. K. Guy and Cal Long, Email to N. J. A. Sloane, Feb 22, 2002.

%D 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 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 Vincenzo Librandi, <a href="/A068555/b068555.txt">Rows n = 0..100, flattened</a>

%H 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 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 Ira Gessel, <a href="https://people.brandeis.edu/~gessel/homepage/slides/nonneg.pdf">Rational functions with nonnegative power series</a>, (slides).

%H Ira Gessel, <a href="https://people.brandeis.edu/~gessel/homepage/papers/superballot.pdf">Super ballot numbers</a>.

%H 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 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 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 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 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 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 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 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 From _Bruno Berselli_, Apr 27 2012: (Start)

%e Triangle begins:

%e        1;

%e        2,    2;

%e        6,    2,    6;

%e       20,    4,    4,  20;

%e       70,   10,    6,  10,  70;

%e      252,   28,   12,  12,  28, 252;

%e      924,   84,   28,  20,  28,  84, 924;

%e     3432,  264,   72,  40,  40,  72, 264, 3432;

%e    12870,  858,  198,  90,  70,  90, 198,  858, 12870;

%e    48620, 2860,  572, 220, 140, 140, 220,  572,  2860, 48620;

%e   184756, 9724, 1716, 572, 308, 252, 308,  572,  1716,  9724, 184756; ...

%e (End)

%e 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 := proc(n,i)

%p     j := n-i ;

%p     (2*i)!*(2*j)!/(i!*j!*(i+j)!) ;

%p end proc: # _R. J. Mathar_, May 31 2016

%t Flatten[ Table[ Table[ (2i)!*(2(n - i))!/(i!*(n - i)!*n!), {i, 0, n}], {n, 0, 9}]]

%o (PARI) a(n,k)=if(n<0 || k<0,0,(2*n)!*(2*k)!/n!/k!/(n+k)!);

%o (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 Apart perhaps from signs, diagonals give A000984, A002420, A078718.

%Y Cf. A007318, A046521, A182073.

%Y Cf. A182411, A082590 (row sums).

%K nonn,tabl,easy,nice

%O 0,2

%A _N. J. A. Sloane_, Mar 23 2002