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a(n) is the smallest number m such that binomial(m,n) is nonzero and is divisible by n!.
1

%I #38 Feb 11 2020 18:55:43

%S 1,4,9,33,28,165,54,1029,40832,31752,28680,2588680,2162700,12996613,

%T 12341252,4516741125,500367376,133207162881,93770874890,7043274506259,

%U 40985291653137,70766492123145,321901427163142,58731756479578128,676814631896875010,6820060161969750025

%N a(n) is the smallest number m such that binomial(m,n) is nonzero and is divisible by n!.

%C a(n) is such that a nontrivial n-symmetric permutation of [1..a(n)] might exist.

%H Bert Dobbelaere, <a href="/A320920/b320920.txt">Table of n, a(n) for n = 1..34</a>

%H Bert Dobbelaere, <a href="/A320920/a320920.py.txt">Python program</a>

%H Tanya Khovanova, <a href="https://blog.tanyakhovanova.com/2018/10/3-symmetric-permutations/#comment-12716">3-Symmetric Permutations</a>

%e The sequence of binomial coefficients C(n,3) starts as: 0, 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, and so on. The smallest nonzero number divisible by 3! is 84, which is C(9,3). Therefore a(3) = 9.

%t a[n_] := Module[{w, m, bc}, {w, m} = {n!, n}; bc[i_] := Binomial[n-1, i] ~Mod~ w; While[True, bc[n] = (bc[n-1] + bc[n]) ~Mod~ w; If[bc[n] == 0, Return[m]]; For[i = n-1, i >= 0, i--, bc[i] = (bc[i-1] + bc[i]) ~Mod~ w]; m++]];

%t Array[a, 12] (* _Jean-François Alcover_, May 31 2019, after _Chai Wah Wu_ *)

%o (Python)

%o from sympy import factorial, binomial

%o def A320920(n):

%o w, m = int(factorial(n)), n

%o bc = [int(binomial(n-1,i)) % w for i in range(n+1)]

%o while True:

%o bc[n] = (bc[n-1]+bc[n]) % w

%o if bc[n] == 0:

%o return m

%o for i in range(n-1,0,-1):

%o bc[i] = (bc[i-1]+bc[i]) % w

%o m += 1 # _Chai Wah Wu_, Oct 25 2018

%Y Cf. A042948, A316775, A320919.

%K nonn

%O 1,2

%A _Tanya Khovanova_, Oct 24 2018

%E a(14)-a(15) from _Alois P. Heinz_, Oct 24 2018

%E a(16)-a(17) from _Chai Wah Wu_, Oct 25 2018

%E a(18)-a(19) from _Giovanni Resta_, Oct 26 2018

%E a(20) from _Giovanni Resta_, Oct 27 2018

%E a(21) and beyond from _Bert Dobbelaere_, Feb 11 2020