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%I #50 Jan 12 2024 11:38:09
%S 1,-4,9,-16,25,-36,49,-64,81,-100,121,-144,169,-196,225,-256,289,-324,
%T 361,-400,441,-484,529,-576,625,-676,729,-784,841,-900,961,-1024,1089,
%U -1156,1225,-1296,1369,-1444,1521,-1600,1681,-1764,1849,-1936,2025,-2116,2209,-2304,2401,-2500
%N a(n) = -(-1)^n * n^2.
%C This sequence is the denominator of (Pi^2)/12 = 1/1-1/4+1/9-1/16+1/25-1/36+... - _Mohammad K. Azarian_, Dec 29 2011
%C Also, circulant determinant of [1,2,...,n,n-1,...,1], i.e., determinant of the (2n-1) X (2n-1) matrix which has this as first row (and also first column), where row k+1 is obtained by cyclically shifting row k one place to the left. - _M. F. Hasler_, Dec 17 2016
%H Vincenzo Librandi, <a href="/A162395/b162395.txt">Table of n, a(n) for n = 1..1000</a>
%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/rfmc.html">Rational Function Multiplicative Coefficients</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-3,-3,-1).
%F Euler transform of length 2 sequence [-4, 3].
%F a(n) is multiplicative with a(2^e) = -(4^e) if e>0, a(p^e) = (p^2)^e if p>2.
%F G.f.: x * (1 - x) / (1 + x)^3.
%F E.g.f.: exp(-x) * (x - x^2).
%F a(n) = a(-n) = -(-1)^n * A000290(n) for all n in Z.
%F Sum_{n>=1} 1/a(n) = Pi^2/12 (A072691). - _Amiram Eldar_, Dec 10 2022
%F Dirichlet g.f.: zeta(s-2)*(1-2^(3-s)). - _Amiram Eldar_, Jan 07 2023
%e G.f. = x - 4*x^2 + 9*x^3 - 16*x^4 + 25*x^5 - 36*x^6 + 49*x^7 - 64*x^8 + 81*x^9 + ...
%t Table[(-1)^(n+1) * n^2, {n, 60}] (* _Vincenzo Librandi_, Feb 15 2013 *)
%o (PARI) {a(n) = -(-1)^n * n^2};
%o (Magma) [(-1)^(n+1) * n^2: n in [1..60]]; // _Vincenzo Librandi_, Feb 15 2013
%Y Cf. A000290, A072691.
%Y For the reversion of this sequence see A263843 (and also A007297).
%K sign,mult,easy
%O 1,2
%A _Michael Somos_, Jul 02 2009
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