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A068555
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Triangle read by rows in which row n contains (2i)!*(2j)!/(i!*j!*(i+j)!) for i+j=n, i=0..n.
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6
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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
(list;
table;
graph;
refs;
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history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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Suggested by R. K. Guy and Cal Long, Feb 22, 2002
Larcombe, Peter J.; French, David R. 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.
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LINKS
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Table of n, a(n) for n=0..56.
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.
Ira Gessel, Rational functions with nonnegative power series, (slides).
Ira Gessel, Super ballot numbers.
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FORMULA
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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.
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
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EXAMPLE
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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; ...
[Bruno Berselli, Apr 27 2012]
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MATHEMATICA
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Flatten[ Table[ Table[ (2i)!*(2(n - i))!/(i!*(n - i)!*n!), {i, 0, n}], {n, 0, 9}]]
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PROG
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(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
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CROSSREFS
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Apart perhaps from signs, diagonals give A000984, A002420, A078718. A007318, A046521, A182073.
Cf. A182411.
Sequence in context: A084426 A187564 A138061 * A167556 A221438 A193322
Adjacent sequences: A068552 A068553 A068554 * A068556 A068557 A068558
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KEYWORD
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nonn,tabl,easy,nice
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AUTHOR
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N. J. A. Sloane, Mar 23 2002
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EXTENSIONS
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More terms from John W. Layman and Robert G. Wilson v, Mar 27 2002
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STATUS
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approved
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