A068553
a(n) = lcm(1,2,...,2*n) / (n*binomial(2*n, n)).
Original entry on oeis.org
1, 1, 1, 3, 2, 5, 15, 7, 28, 126, 30, 165, 198, 143, 1001, 15015, 3640, 884, 7956, 1938, 19380, 203490, 49742, 572033, 980628, 240350, 3124550, 766935, 188370, 2731365, 40970475, 20160075, 4962480, 81880920, 20173560, 353037300
Offset: 1
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[Lcm([1..2*n])/(n*(n+1)*Catalan(n)): n in [1..50]]; // G. C. Greubel, May 04 2023
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Num:= 2: Den:=2: Res:= 1:
for n from 2 to 100 do
Num:= ilcm(Num,2*n-1,2*n);
Den:= Den*(4+2/(n-1));
Res:= Res, Num/Den;
od:
Res; # Robert Israel, Dec 26 2018
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Table[(LCM@@Range[2n])/(n Binomial[2n,n]),{n,40}] (* Harvey P. Dale, Jul 17 2012 *)
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def A068553(n) -> int:
return lcm(range(1,2*n+1))//(n*binomial(2*n,n))
[A068553(n) for n in range(1,51)] # G. C. Greubel, May 04 2023
A268512
Triangle of coefficients c(n,i), 1<=i<=n, such that for each n>=2, c(n,i) are setwise coprime; and for all primes p>2n-1, the sum of (-1)^i*c(n,i)*binomial(i*p,p) is divisible by p^(2n-1).
Original entry on oeis.org
1, 2, 1, 12, 9, 2, 60, 54, 20, 3, 840, 840, 400, 105, 12, 2520, 2700, 1500, 525, 108, 10, 27720, 31185, 19250, 8085, 2268, 385, 30, 360360, 420420, 280280, 133770, 45864, 10780, 1560, 105, 720720, 864864, 611520, 321048, 127008, 36960, 7488, 945, 56, 12252240, 15036840, 11138400, 6297480, 2776032, 942480
Offset: 1
n=1: 1
n=2: 2, 1
n=3: 12, 9, 2
n=4: 60, 54, 20, 3
n=5: 840, 840, 400, 105, 12
...
For all primes p>3, p^3 divides 2 - binomial(2*p,p) (cf. A087754).
For all primes p>5, p^5 divides 12 - 9*binomial(2*p,p) + 2*binomial(3*p,p) (cf. A268589).
For all primes p>7, p^7 divides 60 - 54*binomial(2*p,p) + 20*binomial(3*p,p) - 3*binomial(4*p,p) (cf. A268590).
- R. R. Aidagulov, M. A. Alekseyev. On p-adic approximation of sums of binomial coefficients. Journal of Mathematical Sciences 233:5 (2018), 626-634. doi:10.1007/s10958-018-3948-0; also arXiv, arXiv:1602.02632 [math.NT], 2016-2018.
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a3418[n_] := LCM @@ Range[n];
c[1, 1] = 1; c[n_, i_] := a3418[2(n-1)] Binomial[2n-1, n-i] ((2i-1)/i/ Binomial[2n-1, n]);
Table[c[n, i], {n, 1, 10}, {i, 1, n}] // Flatten (* Jean-François Alcover, Dec 04 2018 *)
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{ A268512(n,i) = lcm(vector(2*(n-1),i,i)) * binomial(2*n-1,n-i) * (2*i-1) / i / binomial(2*n-1,n) }
A048619
a(n) = LCM(binomial(n,0), ..., binomial(n,n)) / binomial(n,floor(n/2)).
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 3, 3, 4, 2, 10, 5, 30, 15, 7, 7, 56, 28, 252, 126, 60, 30, 330, 165, 396, 198, 286, 143, 2002, 1001, 15015, 15015, 7280, 3640, 1768, 884, 15912, 7956, 3876, 1938, 38760, 19380, 406980, 203490, 99484, 49742, 1144066, 572033, 1961256, 980628
Offset: 0
If n=10 then A002944(10)=2520, A001405(10)=252, the quotient a(10)=10.
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[Lcm([1..n+1]) div (Floor((n+3)/2)*Binomial(n+1,Floor((n+3)/2))): n in [0..50]]; // Vincenzo Librandi, Jul 10 2019
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Table[Apply[LCM, Binomial[n, Range[0, n]]]/Binomial[n, Floor[n/2]], {n, 0, 48}] (* Michael De Vlieger, Jun 29 2017 *)
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{A048619(n) = lcm(vector(n+1, i, i)) / binomial(n+1, (n+1)\2) / ((n+2)\2);}
Definition corrected and a(0)=1 prepended by
Max Alekseyev, Oct 23 2015
A263673
a(n) = lcm{1,2,...,n} / binomial(n,floor(n/2)).
Original entry on oeis.org
1, 1, 1, 2, 2, 6, 3, 12, 12, 20, 10, 60, 30, 210, 105, 56, 56, 504, 252, 2520, 1260, 660, 330, 3960, 1980, 5148, 2574, 4004, 2002, 30030, 15015, 240240, 240240, 123760, 61880, 31824, 15912, 302328, 151164, 77520, 38760, 813960, 406980, 8953560, 4476780, 2288132, 1144066, 27457584, 13728792, 49031400
Offset: 0
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a := n -> lcm(seq(k,k=1..n))/binomial(n,iquo(n,2)):
seq(a(n), n=0..49); # Peter Luschny, Oct 23 2015
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Join[{1}, Table[LCM @@ Range[n]/Binomial[n, Floor[n/2]], {n, 1, 50}]] (* or *) Table[Product[Cyclotomic[k, 1], {k, 2, n}]/Binomial[n, Floor[n/2]], {n, 0, 50}] (* G. C. Greubel, Apr 17 2017 *)
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A263673(n) = lcm(vector(n,i,i)) / binomial(n,n\2);
Showing 1-4 of 4 results.
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