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

0,2

Comments

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

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 [math.CO], 2014-2015.

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A030004 A271512 A294473 * A267554 A337434 A328490

Adjacent sequences: A174180 A174181 A174182 * A174184 A174185 A174186

Keyword

nonn,easy,base

Author

Michel Lagneau, Mar 11 2010

Extensions

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

Status

approved