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Palindromes using exactly four different symbols.
8

%I #21 Sep 28 2018 10:01:12

%S 0,0,0,0,0,0,24,24,240,240,1560,1560,8400,8400,40824,40824,186480,

%T 186480,818520,818520,3498000,3498000,14676024,14676024,60780720,

%U 60780720,249401880,249401880,1016542800,1016542800,4123173624,4123173624,16664094960,16664094960

%N Palindromes using exactly four different symbols.

%D M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,9,-9,-26,26,24,-24).

%F a(n) = 4! * Stirling2( [(n+1)/2], 4).

%F G.f.: 24*x^7/((1-x)*(1-2*x)*(1+2*x)*(1-2*x^2)*(1-3*x^2)). - _Colin Barker_, May 05 2012

%F G.f.: k!(x^(2k-1)+x^(2k))/Product_{i=1..k}(1-ix^2), where k=4 is the number of symbols. - _Robert A. Russell_, Sep 25 2018

%t k=4; Table[k! StirlingS2[Ceiling[n/2],k],{n,1,30}] (* _Robert A. Russell_, Sep 25 2018 *)

%o (PARI) a(n) = 4!*stirling((n+1)\2, 4, 2); \\ _Altug Alkan_, Sep 25 2018

%Y Cf. A056450, A000919.

%K nonn,easy

%O 1,7

%A _Marks R. Nester_