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

0,2

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

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.

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

Example

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

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}]]

Prog

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

Crossrefs

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

Cf. A007318, A046521, A182073.

Cf. A182411, A082590 (row sums).

Sequence in context: A138061 A257257 A257251 * A167556 A221438 A327991

Adjacent sequences: A068552 A068553 A068554 * A068556 A068557 A068558

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane, Mar 23 2002

Status

approved