A068555 Triangle read by rows in which row n contains (2i)!*(2j)!/(i!*j!*(i+j)!) for i + j = n, i = 0..n.

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, 28, 84, 924, 3432, 264, 72, 40, 40, 72, 264, 3432, 12870, 858, 198, 90, 70, 90, 198, 858, 12870, 48620, 2860, 572, 220, 140, 140, 220, 572, 2860, 48620, 184756, 9724

Offset: 0

Author

N. J. A. Sloane, Mar 23 2002

Keywords

• nonn
• tabl
• easy
• nice

Comments

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

Examples

			From _Bruno Berselli_, Apr 27 2012: (Start)
Triangle begins:
       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,  28,  84, 924;
    3432,  264,   72,  40,  40,  72, 264, 3432;
   12870,  858,  198,  90,  70,  90, 198,  858, 12870;
   48620, 2860,  572, 220, 140, 140, 220,  572,  2860, 48620;
  184756, 9724, 1716, 572, 308, 252, 308,  572,  1716,  9724, 184756; ...
(End)
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
		

References

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

Links

• Vincenzo Librandi, Rows n = 0..100, flattened
• J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, 2007, arXiv:0709.1977v1 [math.NT]; J. London Math. Soc. (2) 79 (2009), 422-444.
• B. Buca and T. Prosen, Connected correlations, fluctuations and current of magnetization in the steady state of boundary driven XXZ spin chains, arXiv preprint arXiv:1509.04911 [cond-mat.stat-mech], 2015.
• Ira Gessel, Rational functions with nonnegative power series, (slides).
• Ira Gessel, Super ballot numbers.
• Thomas M. Richardson, The Reciprocal Pascal Matrix, arXiv preprint arXiv:1405.6315 [math.CO], 2014.
• Thomas M. Richardson, The Super Patalan Numbers, arXiv preprint arXiv:1410.5880 [math.CO], 2014.
• Thomas M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015) # 15.3.3.
• Thomas M. Richardson, The three 'R's and Dual Riordan Arrays, arXiv:1609.01193 [math.CO], 2016.
• R. Sprugnoli, Riordan array proofs of identities in Gould's book.

Crossrefs

Apart perhaps from signs, diagonals give A000984, A002420, A078718.

Cf. A007318, A046521, A182073.

Cf. A182411, A082590 (row sums).

Programs

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

Maple

A068555 := proc(n,i)
    j := n-i ;
    (2*i)!*(2*j)!/(i!*j!*(i+j)!) ;
end proc: # R. J. Mathar, May 31 2016

Mathematica

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

PARI

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

Formula

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.

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

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