A065352 Smallest m such that C(2m,m) is divisible by (m+n)!/m!.

1, 3, 8, 19, 42, 153, 216, 375, 950, 3565, 4068, 12273, 12274, 31729, 122352, 131023, 458222, 522221, 1046508, 3145451, 6291178, 12320745, 16769000, 56623079, 113246182, 267780069, 469745636, 671088611, 1879015394, 2146959329, 6442418144, 16642932703, 16911433694, 60129279965, 206091288540

Offset: 1

Author

Labos Elemer, Oct 31 2001

Keywords

• nonn

Comments

For n=1 see Catalan numbers A000108.

Heuristically one can observe that a(n) + n + 1 has a 'high' valuation of 2. For n = 17..25 we have 2^8|(a(n) + n + 1). - David A. Corneth, Mar 28 2021

Since (m+n)!/m! = C(m+n,m) * n!, Kummer's theorem implies that A000120(a(n)) >= A007814(n!) = A011371(n) = n - A000120(n), and a(n) >= 2^(n-1). - Max Alekseyev, Sep 24 2024

Examples

			n=4: a(4)=19 means that C(38,19)=35345263800 is divisible by (19+1)(19+2)(19+3)(19+4)=23!/19!=20*21*22*23=215520; the quotient is 166315. Smaller (<19) central binomial coefficients are not divisible by such a product of 4 successive terms; the corresponding quotients for n = 1, 2, 3, 4, 5,... are 1, 1, 13, 166315, 9120910752273999,...
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..100
• David A. Corneth, PARI program

Crossrefs

Cf. A000120, A065344-A065350, A002503, A000108, A000984, A007814, A011371.

Programs

Mathematica

Do[m = 1; While[Not[Divisible[Binomial[2*m,m],(m+n)!/m!]], m++]; Print[m], {n, 1, 16}] (* Vaclav Kotesovec, Sep 05 2019 *)

PARI

\\ See Corneth link

Formula

C(2m, m)=A*((m+1)(m+2)...(m+n-1)(m+n)); a(n) is the smallest such m belonging to n: a(n)=Min(m; Mod(A000984(m), (m+n)!/m!)=0)

Extensions

More terms from Naohiro Nomoto, Apr 21 2002

a(16)-a(17) from Vaclav Kotesovec, Sep 06 2019

a(18)-a(25) from David A. Corneth, Mar 28 2021

a(26)-a(31) from David A. Corneth confirmed and terms a(32) onward added by Max Alekseyev, Sep 24 2024