A174183 - OEIS

%I #19 Dec 02 2018 23:08:37

%S 1,10,20,60,240,1200,7200,50400,403200,3628800,36288000,399168000,

%T 4790016000,62270208000,871782912000,13076743680000,209227898880000,

%U 3556874280960000,64023737057280000,1216451004088320000

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

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

%D 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.

%H Harvey P. Dale, <a href="/A174183/b174183.txt">Table of n, a(n) for n = 0..449</a>

%H Michel Lagneau, <a href="/A174183/a174183.pdf">Proof</a>

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081, 2014

%F p(0)=1, and p(k) = 10*k ! for k >=1.

%e x(0)= 0.C(1,0)C(2,0)C(3,0) ... = 0.11111111111... and p(0)=1 ;

%e x(1)= 0.C(1,1)C(2,1)C(3,1) ... = 0.12345678901234... and p(1) = 10 ;

%e x(2)= 0.C(2,2)C(3,2)C(4,2) ... = 0.13605186556815063100 13605186556815063100... and p(2)=20.

%p for a from 0 to 40 do:u:=10*a!:print(u):od:

%t Join[{1},Array[10#!&,20]] (* _Harvey P. Dale_, Feb 18 2018 *)

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

%K nonn,base

%O 0,2

%A _Michel Lagneau_, Mar 11 2010

%E Additional comments, and errors in examples corrected by _Michel Lagneau_, May 07 2010