A174183 a(n) is the period k such that binomial(m, n) (mod 10) = binomial(m + k, n) (mod 10).

1, 10, 20, 60, 240, 1200, 7200, 50400, 403200, 3628800, 36288000, 399168000, 4790016000, 62270208000, 871782912000, 13076743680000, 209227898880000, 3556874280960000, 64023737057280000, 1216451004088320000

Offset: 0

Author

Michel Lagneau, Mar 11 2010

Keywords

• nonn
• easy
• base

Comments

a(n) is the period (mod 10) of the numbers in each column n of Pascal's triangle.

Examples

			x(0)= 0.C(1,0)C(2,0)C(3,0) ... = 0.11111111111... and p(0)=1 ;
x(1)= 0.C(1,1)C(2,1)C(3,1) ... = 0.12345678901234... and p(1) = 10 ;
x(2)= 0.C(2,2)C(3,2)C(4,2) ... = 0.13605186556815063100 13605186556815063100... and p(2)=20.
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..449
• Michel Lagneau, Proof
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

Crossrefs

Cf. A000142, A002415, A007318, A002024, A000096, A000124, A002378, A000292, A000330, A055998, A055999, A056000, A056115, A056119, A056121, A056126, A051942, A101859, A001477.

Programs

Mathematica

Join[{1},Array[10#!&,20]] (* Harvey P. Dale, Feb 18 2018 *)

Python

from math import factorial
def A174183(n): return 10*factorial(n) if n else 1 # Chai Wah Wu, Aug 07 2025

Formula

a(0)=1, and a(n) = 10 * n! for n >= 1.

Extensions

Additional comments, and errors in examples corrected by Michel Lagneau, May 07 2010