A126695 -id:A126695 - cp's OEIS frontend

A130332 Integer values of n!!/sum(i=0..n,n), with n>=1.

1, 1, 21, 1485, 6144, 225225, 17694720, 59520825, 6539968512, 24325703325, 145332633600, 14230536445125, 2596962041856000, 11288163762500625, 78354054748569600, 11665426077721040625, 86068915523813376000

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, May 24 2007

After the ninth term all the other numbers end in 0 or 5.

			5!! = 5*3*1 = 15; 5+4+3+2+1 = 15; 15/15 = 1.
13!! = 13*11*9*7*5*3*1 = 135135; 13+12+11+10+9+8+7+6+5+4+3+2+1 = 91; 135135/91 = 1485.

Harvey P. Dale, Table of n, a(n) for n = 0..466

Cf. A130318, A126695, A108552.

P:=proc(n) local a,i,j,k,w; for i from 1 by 1 to n do k:=i; w:=i-2; while w>0 do k:=k*w; w:=w-2; od; j:=sum('w','w'=1..i); a:=k/j; if trunc(a)=a then print(a) fi; od; end: P(100);

Select[Table[n!!/((n(n+1))/2),{n,50}],IntegerQ] (* Harvey P. Dale, Jul 24 2019 *)

A370601 a(n) is the number of integer values of the function F_2n(x) = Product_{i=0..2n-1} (x + i) / Sum_{i=0..2n-1} (x + i).

Original entry on oeis.org

0, 1, 3, 23, 37, 156, 371, 1207, 2826, 8738, 14839, 60738, 80177, 148702, 466545, 1673039, 2077633, 4771287, 10665251, 26790730, 72170979, 212182718, 248771227, 1074691776, 1488526850, 2533202074, 8444006973, 13950326222, 18313391221, 74263958970, 164820521219

Offset: 1

Vladimir Letsko, Feb 23 2024

nonn

If k is an odd prime then F_k(x) = Product_{i=0..k} (x + i)/ Sum_{i=0..k} (x + i) is not an integer if and only if x==(k+1)/2 (mod k). If k is odd but not prime, then F_k(x) is an integer for all positive integers x. On the other hand, for every even k, there is only a finite number of integer values of F_k(x).

			a(3) = 3 because F_6(x) has exactly 3 integer values: F_6(5) = 3360, F_6(10) = 48048, and F_6(35) = 12282816.

Vladimir Letsko, Math. vertical, YouTube video, 2023 (in Russian).

Cf. A000005, A001818, A126695.

a := proc(n) local d; d := doublefactorial(2*n-1)^2; numtheory[tau](d/igcd(n, d)) - n end: seq(a(n), n = 1..31);

Table[DivisorSigma[0, (2*n - 1)!!^2/GCD[(2*n - 1)!!^2, n]] - n, {n, 1, 30}] (* Vaclav Kotesovec, Feb 23 2024 *)

a(n) = tau(d/gcd(d, n)) - n, where d = ((2*n-1)!!)^2 and tau(k) is the number of divisors of the positive integer k.

cp's OEIS Frontend

A130332 Integer values of n!!/sum(i=0..n,n), with n>=1.

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