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A062024 a(n) = ((n+1)^n + (n-1)^n)/2.

%I A062024 #39 Dec 07 2024 13:39:29
%S A062024 1,1,5,36,353,4400,66637,1188544,24405761,567108864,14712104501,
%T A062024 421504185344,13218256749601,450353989316608,16565151205544957,
%U A062024 654244800082329600,27614800115689879553,1240529732459024678912,59095217374989483261925,2975557672677668838178816
%N A062024 a(n) = ((n+1)^n + (n-1)^n)/2.
%C A062024 Let b(n) = A302583(n) = ((n+1)^n - (n-1)^n)/2 = 0, 1, 4, 28, 272, ... then lim_{n -> infinity} b(n)/a(n) = tanh(1) = 0.76159415... . - _Thomas Ordowski_, Dec 06 2012
%C A062024 Obviously, a(n) is always odd number for even n. - _Altug Alkan_, Sep 28 2015
%H A062024 Harry J. Smith, <a href="/A062024/b062024.txt">Table of n, a(n) for n = 0..100</a>
%F A062024 a(n) = n! * [x^n] exp(n*x)*cosh(x). - _Ilya Gutkovskiy_, Apr 10 2018
%e A062024 a(3) = (4^3 + 2^3)/2 = 36.
%p A062024 A062024:=n->((n+1)^n + (n-1)^n)/2; seq(A062024(n), n=0..20); # _Wesley Ivan Hurt_, Nov 13 2013
%t A062024 a[n_]:=((n-1)^n+(n+1)^n)/2; a[Range[0, 20]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 07 2010; modified by _G. C. Greubel_, Jan 03 2020 *)
%t A062024 Table[((n+1)^n + (n-1)^n)/2, {n,0,20}] (* _Vincenzo Librandi_, Sep 28 2015 *)
%o A062024 (PARI) a(n) = { ((n + 1)^n + (n - 1)^n)/2 } \\ _Harry J. Smith_, Jul 29 2009
%o A062024 (Magma) [((n+1)^n + (n-1)^n)/2: n in [0..20]]; // _Vincenzo Librandi_, Sep 28 2015
%o A062024 (Sage) [((n+1)^n + (n-1)^n)/2 for n in (0..20)] # _G. C. Greubel_, Jan 03 2020
%o A062024 (GAP) List([0..20], n-> ((n+1)^n + (n-1)^n)/2); # _G. C. Greubel_, Jan 03 2020
%Y A062024 Cf. A302583.
%K A062024 nonn
%O A062024 0,3
%A A062024 _Amarnath Murthy_, Jun 02 2001
%E A062024 More terms from Larry Reeves (larryr(AT)acm.org) and _Jason Earls_, Jun 06 2001
%E A062024 Offset changed from 1 to 0 by _Harry J. Smith_, Jul 29 2009
%E A062024 a(18)-a(19) from _Vincenzo Librandi_, Sep 28 2015