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

%I A174183 #25 Aug 07 2025 14:10:27
%S A174183 1,10,20,60,240,1200,7200,50400,403200,3628800,36288000,399168000,
%T A174183 4790016000,62270208000,871782912000,13076743680000,209227898880000,
%U A174183 3556874280960000,64023737057280000,1216451004088320000
%N A174183 a(n) is the period k such that binomial(m, n) (mod 10) = binomial(m + k, n) (mod 10).
%C A174183 a(n) is the period (mod 10) of the numbers in each column n of Pascal's triangle.
%D A174183 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 A174183 Harvey P. Dale, <a href="/A174183/b174183.txt">Table of n, a(n) for n = 0..449</a>
%H A174183 Michel Lagneau, <a href="/A174183/a174183.pdf">Proof</a>
%H A174183 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 A174183 a(0)=1, and a(n) = 10 * n! for n >= 1.
%e A174183 x(0)= 0.C(1,0)C(2,0)C(3,0) ... = 0.11111111111... and p(0)=1 ;
%e A174183 x(1)= 0.C(1,1)C(2,1)C(3,1) ... = 0.12345678901234... and p(1) = 10 ;
%e A174183 x(2)= 0.C(2,2)C(3,2)C(4,2) ... = 0.13605186556815063100 13605186556815063100... and p(2)=20.
%t A174183 Join[{1},Array[10#!&,20]] (* _Harvey P. Dale_, Feb 18 2018 *)
%o A174183 (Python)
%o A174183 from math import factorial
%o A174183 def A174183(n): return 10*factorial(n) if n else 1 # _Chai Wah Wu_, Aug 07 2025
%Y A174183 Cf. A000142, A002415, A007318, A002024, A000096, A000124, A002378, A000292, A000330, A055998, A055999, A056000, A056115, A056119, A056121, A056126, A051942, A101859, A001477.
%K A174183 nonn,easy,base
%O A174183 0,2
%A A174183 _Michel Lagneau_, Mar 11 2010
%E A174183 Additional comments, and errors in examples corrected by _Michel Lagneau_, May 07 2010