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A182971 Triangle read by rows: coefficients in expansion of Q(n) = (x-n^2)*(x-(n-2)^2)*(x-(n-4)^2)*...*(x-(1 or 2)^2), highest powers first.

%I A182971 #20 Aug 29 2012 07:25:12
%S A182971 1,1,-1,1,-4,1,-10,9,1,-20,64,1,-35,259,-225,1,-56,784,-2304,1,-84,
%T A182971 1974,-12916,11025,1,-120,4368,-52480,147456,1,-165,8778,-172810,
%U A182971 1057221,-893025,1,-220,16368,-489280,5395456,-14745600,1,-286,28743,-1234948,21967231,-128816766,108056025,1,-364,48048,-2846272,75851776,-791691264,2123366400
%N A182971 Triangle read by rows: coefficients in expansion of Q(n) = (x-n^2)*(x-(n-2)^2)*(x-(n-4)^2)*...*(x-(1 or 2)^2), highest powers first.
%C A182971 These are scaled versions of the central factorial numbers in A008955 and A008956.
%C A182971 Even-indexed rows give A182867, odd-indexed rows give A008956.
%C A182971 A121408 is an unsigned and aerated version of the row reverse of this triangle. - _Peter Bala_, Aug 29 2012
%F A182971 For n even, let Q(n) = Product_{i=1..n/2} (x - (2*i)^2) and for n odd let Q(n) = Product_{i=0..(n-1)/2} (x - (2i+1)^2). n-th row of triangle gives coefficients in expansion of Q(n).
%e A182971 Triangle begins:
%e A182971 1
%e A182971 1, -1
%e A182971 1, -4
%e A182971 1, -10, 9
%e A182971 1, -20, 64
%e A182971 1, -35, 259, -225
%e A182971 1, -56, 784, -2304
%e A182971 1, -84, 1974, -12916, 11025
%e A182971 1, -120, 4368, -52480, 147456
%e A182971 1, -165, 8778, -172810, 1057221, -893025
%e A182971 1, -220, 16368, -489280, 5395456, -14745600
%e A182971 ...
%e A182971 E.g. for n=5 Q(5) = (x-1^2)*(x-3^2)*(x-5^2) = x^3-35*x^2+259*x-225.
%p A182971 Q:= n -> if n mod 2 = 0 then sort(expand(mul(x-4*i^2,i=1..n/2)));
%p A182971 else sort(expand(mul(x-(2*i+1)^2,i=0..(n-1)/2))); fi;
%p A182971 for n from 0 to 12 do
%p A182971 t1:=eval(Q(n)); t1d:=degree(t1);
%p A182971 t12:=y^t1d*subs(x=1/y,t1); t2:=seriestolist(series(t12,y,20));
%p A182971 lprint(t2);
%p A182971 od:
%Y A182971 Even-indexed rows give A182867, odd-indexed rows give A008956.
%Y A182971 Column 1,4,10,20, ... is A000292. The next two columns give A181888, A184878. The last diagonal is A184877.
%Y A182971 Cf. A008955, A008956. A121408.
%K A182971 sign,tabf
%O A182971 0,5
%A A182971 _N. J. A. Sloane_, Feb 01 2011