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%I #15 Sep 08 2022 08:45:13
%S 1,2,2,3,6,1,4,12,8,12,5,20,5,60,8,6,30,8,20,36,240,7,42,35,420,504,
%T 560,180,8,56,16,840,224,6720,1152,1120,9,72,21,504,432,20160,4320,
%U 90720,8064,10,90,80,2520,756,3360,86400,453600,67200,725760,11,110,33,3960
%N In the triangle shown below the n-th row contains n rational numbers n/1, {n*(n-1)}/{n +(n-1)}, {(n)*(n-1)*(n-2)}/{n +(n-1)+(n-2)}, ..., the last term being 2*(n-1)!/(n+1). Sequence gives the numerators in each row.
%H G. C. Greubel, <a href="/A093422/b093422.txt">Rows n=0..100 of triangle, flattened</a>
%F A093422(n,m)/A093423(n,m) = 2*binomial(n,m)*(m-1)!/(2*n-m+1) for 2 <= m < n. A093422(n,1)/A093423(n,1)= n. - _R. J. Mathar_, Apr 28 2007
%e Triangle of fractions starts
%e 1,
%e 2, 2/3,
%e 3, 6/5, 1,
%e 4, 12/7, 8/3, 12/5,
%e 5, 20/9, 5, 60/7, 8,
%e 6, 30/11, 8, 20, 36, 240/7,
%e 7, 42/13, 35/3, 420/11, 504/5, 560/3, 180,
%e 8, 56/15, 16, 840/13, 224, 6720/11, 1152, 1120,
%e 9, 72/17, 21, 504/5, 432, 20160/13, 4320, 90720/11, 8064,
%p A09342x := proc(n,m) local i,N,D ; N := n ; if m = 1 then D := 1 ; else D := n ; end ; for i from 1 to m-1 do N := N*(n-i) ; D := D+n-i ; od ; simplify(N/D) ; end: A093422 := proc(n,m) numer(A09342x(n,m)) ; end: for n from 1 to 12 do for m from 1 to n do printf("%d, ",A093422(n,m)) ; od ; od ; # _R. J. Mathar_, Apr 28 2007
%t Table[If[k == 1, n, Numerator[2*Binomial[n,k]*(k-1)!/(2*n-k+1)]], {n,1,30}, {k,1,n}]//Flatten (* _G. C. Greubel_, Sep 01 2018 *)
%o (PARI) for(n=1,10, for(k=1,n, print1(if(k==1, n, denominator(2*binomial(n,k)*(k-1)!/(2*n-k+1))), ", "))) \\ _G. C. Greubel_, Sep 01 2018
%o (Magma) /* as a triangle */ [[k le 1 select n else Numerator(2*Binomial(n,k)*Factorial(k-1)/(2*n-k+1)): k in [1..n]]: n in [1..10]]; // _G. C. Greubel_, Sep 01 2018
%Y Cf. A093412, A093413, A093414, A093415, A093417, A093418, A093419, A093420, A093421, A093423.
%K easy,frac,nonn,tabl
%O 1,2
%A _Amarnath Murthy_, Mar 30 2004
%E More terms from _R. J. Mathar_, Apr 28 2007
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