A093422 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.
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, 560, 180, 8, 56, 16, 840, 224, 6720, 1152, 1120, 9, 72, 21, 504, 432, 20160, 4320, 90720, 8064, 10, 90, 80, 2520, 756, 3360, 86400, 453600, 67200, 725760, 11, 110, 33, 3960
Offset: 1
Examples
Triangle of fractions starts 1, 2, 2/3, 3, 6/5, 1, 4, 12/7, 8/3, 12/5, 5, 20/9, 5, 60/7, 8, 6, 30/11, 8, 20, 36, 240/7, 7, 42/13, 35/3, 420/11, 504/5, 560/3, 180, 8, 56/15, 16, 840/13, 224, 6720/11, 1152, 1120, 9, 72/17, 21, 504/5, 432, 20160/13, 4320, 90720/11, 8064,
Links
- G. C. Greubel, Rows n=0..100 of triangle, flattened
Crossrefs
Programs
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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
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Maple
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
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Mathematica
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 *) -
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
Formula
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
Extensions
More terms from R. J. Mathar, Apr 28 2007