Search: id:A035287
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%I A035287
%S A035287 0,4,36,144,400,900,1764,3136,5184,8100,12100,17424,24336,33124,44100,
%T A035287 57600,73984,93636,116964,144400,176400,213444,256036,304704,360000,
%U A035287 422500,492804,571536,659344,756900,864900,984064,1115136,1258884
%N A035287 Number of ways to place a non-attacking white and black rook on n X n 
               chessboard.
%C A035287 a(n) is equal to the number of functions f:{1,2,3,4}->{1,2,...,n} such 
               that for fixed different x_1, x_2 in {1,2,3,4} and fixed y_1, y_2 
               in {1,2,...,n} we have f(x_1)<>y_1 and f(x_2)<>y_2. - Milan R. Janjic 
               (agnus(AT)blic.net), Apr 17 2007
%C A035287 a(n+1)=4, A061038(1), A061040(1), A061042(1), A061044(1), A061046(1), 
               A061048(1), A061050(1);after 4 (which is linked to Lyman by "companion" 
               of A005563, A000290(n+1)?), from Balmer, Paschen, Brackett, Pfund, 
               Humphreys, Hansen-Strong, .., s-th spectra of hydrogen. Corresponding 
               numerators or "companions" are A144396. [From Paul Curtz (bpcrtz(AT)free.fr), 
               Nov 05 2008]
%C A035287 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 21 
               2009: (Start)
%C A035287 The third differences of certain values of the hypergeometric function 
               3F2 lead to this sequence i.e. 3F2([1,n+1,n+1], [n+2,n+2], z=1) - 
               3*3F2([1,n+2,n+2], [n+3,n+3], z=1) + 3*3F2([1,n+3,n+3], [n+4,n+4], 
               z=1) - 3F2([1,n+4,n+4], [n+5,n+5], z=1) = (1/((n+2)*(n+3)))^2 with 
               n = -1, 0, 1,2, .. . See also A162990.
%C A035287 (End)
%H A035287 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas 
               for Some Functions on Finite Sets</a>
%H A035287 Wolfram Research, <a href="http://functions.wolfram.com/webMathematica/
               FunctionEvaluation.jsp?name=Hypergeometric3F2">Hypergeometric Function 
               3F2</a>, The Wolfram Functions site. [From Johannes W. Meijer (meijgia(AT)hotmail.com), 
               Jul 21 2009]
%F A035287 Diagonal of [A132710]*Transpose[A132710]. - Tom Copeland (tcjpn(AT)msn.com), 
               Nov 20 2007
%F A035287 a(n) = n^2 (n-1)^2
%F A035287 a(n) = A002378(n)^2. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 11 2006
%F A035287 Contribution from Stephen Crowley (crow(AT)crowlogic.net), Jul 19 2009: 
               (Start)
%F A035287 a(n)=limit(n!*(2*n+1)/(diff(1+polylog(2,x)-polylog(2,x)/x, x$n),x=0) 
               and int(1+polylog(2,x)-polylog(2,x)/x,x=0..1)=Zeta(2)-Zeta(3)
%F A035287 sum((n^2*(n-1)^2)^(-1),n=2..infinity)=2*Zeta(2)-3 (End)
%p A035287 seq((numbperm (n,2))^2, n=1..34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 26 2007
%p A035287 a:=n->sum(sum(n^2, j=0..n),k=0..n): seq(a(n), n=0..33); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), May 09 2007
%p A035287 a:=n->sum(sum (2*binomial(n,2),j=2..n),k=1..n): seq(a(n),n=1..34); - 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 02 2007
%p A035287 map(f->if(f=0,0,denom(f)), PolynomialTools[CoefficientList](convert(series(1+polylog(2, 
               x)-polylog(2, x)/x, x, 20), polynom), x)) [From Stephen Crowley (crow(AT)crowlogic.net), 
               Jul 19 2009]
%t A035287 lst={};Do[s0=n^2;s1=(n+1)^2;AppendTo[lst,s1*s0],{n,0,6!}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Feb 19 2009]
%o A035287 (Other) sage: [n^2*(n-1)^2 for n in xrange(1, 35)] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Dec 03 2009]
%Y A035287 Cf. A002378.
%Y A035287 Sequence in context: A045490 A060783 A125756 this_sequence A083223 A102263 
               A103931
%Y A035287 Adjacent sequences: A035284 A035285 A035286 this_sequence A035288 A035289 
               A035290
%K A035287 nonn,new
%O A035287 1,2
%A A035287 Erich Friedman (erich.friedman(AT)stetson.edu)

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